在发送地址里填写IPV6地址，而不是IPV4地址。

先布置窗口元素：

namespace Demo
{
    partial class Form1
    {
        /// <summary>
        /// 必需的设计器变量。
        /// </summary>
        private System.ComponentModel.IContainer components = null;

        /// <summary>
        /// 清理所有正在使用的资源。
        /// </summary>
        /// <param name="disposing">如果应释放托管资源，为 true；否则为 false。</param>
        protected override void Dispose(bool disposing)
        {
            if (disposing && (components != null))
            {
                components.Dispose();
            }
            base.Dispose(disposing);
        }

        #region Windows 窗体设计器生成的代码

        /// <summary>
        /// 设计器支持所需的方法 - 不要修改
        /// 使用代码编辑器修改此方法的内容。
        /// </summary>
        private void InitializeComponent()
        {
            this.components = new System.ComponentModel.Container();
            this.splitContainer1 = new System.Windows.Forms.SplitContainer();
            this.rtbContent = new System.Windows.Forms.RichTextBox();
            this.splitContainer2 = new System.Windows.Forms.SplitContainer();
            this.rtbSend = new System.Windows.Forms.RichTextBox();
            this.txtName = new System.Windows.Forms.TextBox();
            this.label2 = new System.Windows.Forms.Label();
            this.txtIP = new System.Windows.Forms.TextBox();
            this.label1 = new System.Windows.Forms.Label();
            this.btn_Clear = new System.Windows.Forms.Button();
            this.btn_Send = new System.Windows.Forms.Button();
            this.btn_Close = new System.Windows.Forms.Button();
            this.timer1 = new System.Windows.Forms.Timer(this.components);
            ((System.ComponentModel.ISupportInitialize)(this.splitContainer1)).BeginInit();
            this.splitContainer1.Panel1.SuspendLayout();
            this.splitContainer1.Panel2.SuspendLayout();
            this.splitContainer1.SuspendLayout();
            ((System.ComponentModel.ISupportInitialize)(this.splitContainer2)).BeginInit();
            this.splitContainer2.Panel1.SuspendLayout();
            this.splitContainer2.Panel2.SuspendLayout();
            this.splitContainer2.SuspendLayout();
            this.SuspendLayout();
            // 
            // splitContainer1
            // 
            this.splitContainer1.BorderStyle = System.Windows.Forms.BorderStyle.FixedSingle;
            this.splitContainer1.Location = new System.Drawing.Point(3, 4);
            this.splitContainer1.Name = "splitContainer1";
            this.splitContainer1.Orientation = System.Windows.Forms.Orientation.Horizontal;
            // 
            // splitContainer1.Panel1
            // 
            this.splitContainer1.Panel1.Controls.Add(this.rtbContent);
            // 
            // splitContainer1.Panel2
            // 
            this.splitContainer1.Panel2.Controls.Add(this.splitContainer2);
            this.splitContainer1.Size = new System.Drawing.Size(484, 400);
            this.splitContainer1.SplitterDistance = 264;
            this.splitContainer1.SplitterWidth = 3;
            this.splitContainer1.TabIndex = 0;
            // 
            // rtbContent
            // 
            this.rtbContent.BackColor = System.Drawing.Color.White;
            this.rtbContent.BorderStyle = System.Windows.Forms.BorderStyle.None;
            this.rtbContent.Dock = System.Windows.Forms.DockStyle.Fill;
            this.rtbContent.Location = new System.Drawing.Point(0, 0);
            this.rtbContent.Name = "rtbContent";
            this.rtbContent.ReadOnly = true;
            this.rtbContent.Size = new System.Drawing.Size(482, 262);
            this.rtbContent.TabIndex = 0;
            this.rtbContent.Text = "";
            // 
            // splitContainer2
            // 
            this.splitContainer2.BorderStyle = System.Windows.Forms.BorderStyle.FixedSingle;
            this.splitContainer2.Dock = System.Windows.Forms.DockStyle.Fill;
            this.splitContainer2.Location = new System.Drawing.Point(0, 0);
            this.splitContainer2.Name = "splitContainer2";
            this.splitContainer2.Orientation = System.Windows.Forms.Orientation.Horizontal;
            // 
            // splitContainer2.Panel1
            // 
            this.splitContainer2.Panel1.Controls.Add(this.rtbSend);
            // 
            // splitContainer2.Panel2
            // 
            this.splitContainer2.Panel2.Controls.Add(this.txtName);
            this.splitContainer2.Panel2.Controls.Add(this.label2);
            this.splitContainer2.Panel2.Controls.Add(this.txtIP);
            this.splitContainer2.Panel2.Controls.Add(this.label1);
            this.splitContainer2.Panel2.Controls.Add(this.btn_Clear);
            this.splitContainer2.Panel2.Controls.Add(this.btn_Send);
            this.splitContainer2.Panel2.Controls.Add(this.btn_Close);
            this.splitContainer2.Size = new System.Drawing.Size(484, 133);
            this.splitContainer2.SplitterDistance = 101;
            this.splitContainer2.SplitterWidth = 1;
            this.splitContainer2.TabIndex = 0;
            // 
            // rtbSend
            // 
            this.rtbSend.BorderStyle = System.Windows.Forms.BorderStyle.None;
            this.rtbSend.Dock = System.Windows.Forms.DockStyle.Fill;
            this.rtbSend.Location = new System.Drawing.Point(0, 0);
            this.rtbSend.Name = "rtbSend";
            this.rtbSend.Size = new System.Drawing.Size(482, 99);
            this.rtbSend.TabIndex = 0;
            this.rtbSend.Text = "";
            this.rtbSend.KeyPress += new System.Windows.Forms.KeyPressEventHandler(this.rtbSend_KeyPress);
            // 
            // txtName
            // 
            this.txtName.Location = new System.Drawing.Point(240, 5);
            this.txtName.Name = "txtName";
            this.txtName.Size = new System.Drawing.Size(100, 21);
            this.txtName.TabIndex = 1;
            // 
            // label2
            // 
            this.label2.AutoSize = true;
            this.label2.Location = new System.Drawing.Point(200, 11);
            this.label2.Name = "label2";
            this.label2.Size = new System.Drawing.Size(41, 12);
            this.label2.TabIndex = 4;
            this.label2.Text = "昵称：";
            // 
            // txtIP
            // 
            this.txtIP.Location = new System.Drawing.Point(67, 6);
            this.txtIP.Name = "txtIP";
            this.txtIP.Size = new System.Drawing.Size(127, 21);
            this.txtIP.TabIndex = 0;
            // 
            // label1
            // 
            this.label1.AutoSize = true;
            this.label1.Location = new System.Drawing.Point(8, 10);
            this.label1.Name = "label1";
            this.label1.Size = new System.Drawing.Size(65, 12);
            this.label1.TabIndex = 1;
            this.label1.Text = "对方主机：";
            // 
            // btn_Clear
            // 
            this.btn_Clear.Location = new System.Drawing.Point(346, 3);
            this.btn_Clear.Name = "btn_Clear";
            this.btn_Clear.Size = new System.Drawing.Size(39, 23);
            this.btn_Clear.TabIndex = 3;
            this.btn_Clear.Text = "清屏";
            this.btn_Clear.UseVisualStyleBackColor = true;
            this.btn_Clear.Click += new System.EventHandler(this.btn_Clear_Click);
            // 
            // btn_Send
            // 
            this.btn_Send.Location = new System.Drawing.Point(391, 3);
            this.btn_Send.Name = "btn_Send";
            this.btn_Send.Size = new System.Drawing.Size(38, 23);
            this.btn_Send.TabIndex = 2;
            this.btn_Send.Text = "发送";
            this.btn_Send.UseVisualStyleBackColor = true;
            this.btn_Send.Click += new System.EventHandler(this.btn_Send_Click);
            // 
            // btn_Close
            // 
            this.btn_Close.Location = new System.Drawing.Point(435, 3);
            this.btn_Close.Name = "btn_Close";
            this.btn_Close.Size = new System.Drawing.Size(39, 23);
            this.btn_Close.TabIndex = 4;
            this.btn_Close.Text = "关闭";
            this.btn_Close.UseVisualStyleBackColor = true;
            this.btn_Close.Click += new System.EventHandler(this.btn_Close_Click);
            // 
            // timer1
            // 
            this.timer1.Interval = 1000;
            this.timer1.Tick += new System.EventHandler(this.timer1_Tick);
            // 
            // Form1
            // 
            this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 12F);
            this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
            this.BackColor = System.Drawing.Color.FromArgb(((int)(((byte)(223)))), ((int)(((byte)(233)))), ((int)(((byte)(243)))));
            this.ClientSize = new System.Drawing.Size(490, 407);
            this.Controls.Add(this.splitContainer1);
            this.FormBorderStyle = System.Windows.Forms.FormBorderStyle.FixedSingle;
            this.MaximizeBox = false;
            this.Name = "Form1";
            this.StartPosition = System.Windows.Forms.FormStartPosition.CenterScreen;
            this.Text = "点对点聊天";
            this.FormClosed += new System.Windows.Forms.FormClosedEventHandler(this.frmMain_FormClosed);
            this.Load += new System.EventHandler(this.frmMain_Load);
            this.splitContainer1.Panel1.ResumeLayout(false);
            this.splitContainer1.Panel2.ResumeLayout(false);
            ((System.ComponentModel.ISupportInitialize)(this.splitContainer1)).EndInit();
            this.splitContainer1.ResumeLayout(false);
            this.splitContainer2.Panel1.ResumeLayout(false);
            this.splitContainer2.Panel2.ResumeLayout(false);
            this.splitContainer2.Panel2.PerformLayout();
            ((System.ComponentModel.ISupportInitialize)(this.splitContainer2)).EndInit();
            this.splitContainer2.ResumeLayout(false);
            this.ResumeLayout(false);

        }

        #endregion

        private System.Windows.Forms.SplitContainer splitContainer1;
        private System.Windows.Forms.RichTextBox rtbContent;
        private System.Windows.Forms.SplitContainer splitContainer2;
        private System.Windows.Forms.RichTextBox rtbSend;
        private System.Windows.Forms.Button btn_Clear;
        private System.Windows.Forms.Button btn_Send;
        private System.Windows.Forms.Button btn_Close;
        private System.Windows.Forms.TextBox txtName;
        private System.Windows.Forms.Label label2;
        private System.Windows.Forms.TextBox txtIP;
        private System.Windows.Forms.Label label1;
        private System.Windows.Forms.Timer timer1;
    }
}

 再上代码，代码简单，不要过多解释。

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.IO;
using System.Linq;
using System.Net;
using System.Net.Sockets;
using System.Text;
using System.Threading;
using System.Threading.Tasks;
using System.Windows.Forms;

namespace Demo
{
    public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();
        }
        private Thread td;
        private TcpListener tcpListener;
        private static string message = "";

        private void frmMain_Load(object sender, EventArgs e)
        {
            td = new Thread(new ThreadStart(this.StartListen));
            td.Start();
            timer1.Start();
            IPAddress[] ip = Dns.GetHostAddresses(Dns.GetHostName());
            txtIP.Text = ip[0].ToString();
        }



        private void StartListen()
        {
            message = "";
            IPAddress[] ip = Dns.GetHostAddresses(Dns.GetHostName());
            tcpListener = new TcpListener(ip[0],888);
            tcpListener.Start();
            while (true)
            {
                TcpClient tclient = tcpListener.AcceptTcpClient();  //接受连接请求
                NetworkStream nstream = tclient.GetStream();        //获取数据流
                byte[] mbyte = new byte[1024];                      //建立缓存
                int i = nstream.Read(mbyte, 0, mbyte.Length);       //将数据流写入缓存
                message = Encoding.Default.GetString(mbyte, 0, i);
            }
        }

        private void frmMain_FormClosed(object sender, FormClosedEventArgs e)
        {
            if (this.tcpListener != null)
            {
                tcpListener.Stop();
            }
            if (td != null)
            {
                if (td.ThreadState == ThreadState.Running)
                {
                    td.Abort();
                }
            }
        }

        private void btn_Clear_Click(object sender, EventArgs e)
        {
            rtbContent.Clear();
        }

        private void rtbSend_KeyPress(object sender, KeyPressEventArgs e)
        {
            if (e.KeyChar == '\r')
            {
                btn_Send_Click(sender, e);
            }
        }

        private void timer1_Tick(object sender, EventArgs e)
        {
            if (message != "")
            {
                rtbContent.AppendText(message);
                rtbContent.ScrollToCaret();
                message = "";
            }
        }

        private void btn_Close_Click(object sender, EventArgs e)
        {
            Application.Exit();
        }

        private void btn_Send_Click(object sender, EventArgs e)
        {
            try
            {
                IPAddress[] ip = Dns.GetHostAddresses(Dns.GetHostName());
                string strmsg = " " + txtName.Text + "(" + ip[0].ToString() + ") " + DateTime.Now.ToLongTimeString() + "\n" + "  " + this.rtbSend.Text + "\n";
                TcpClient client = new TcpClient(txtIP.Text, 888);
                NetworkStream netstream = client.GetStream();
                StreamWriter wstream = new StreamWriter(netstream, Encoding.Default);
                wstream.Write(strmsg);
                wstream.Flush();
                wstream.Close();
                client.Close();
                rtbContent.AppendText(strmsg);
                rtbContent.ScrollToCaret();
                rtbSend.Clear();
            }
            catch (Exception ex)
            {
                MessageBox.Show(ex.Message);
            }

        }
    }
}