Credit: Michael Lipp (Microsoft)

Here's some sample code that sorts a listview when you click on a column
header:

using System;
using System.Drawing;
using System.Windows.Forms;
using System.Collections;

public class Test : System.Windows.Forms.Form
{
    ListView lv;

    ListViewItem MakeItem(char c)
    {
        string s = "";
        for (int n = 0; n < 5; n++)
            s += c.ToString();

        ListViewItem item = new ListViewItem(new string[] {
            "1 " + s,
            "2 " + s,
            "3 " + s
        });
        return item;
    }

    protected override void OnLoad(EventArgs e)
    {
        this.Text = "myform";

        this.lv = new ListView();
        this.lv.Parent = this;
        this.lv.Bounds = this.ClientRectangle;
        this.lv.View = View.Details;
        this.lv.ColumnClick += new 
ColumnClickEventHandler(this.lv_ColumnClick);

        for(int n = 0; n < 3; n++)
            lv.Columns.Add(new myColumnHeader("column" + n, 70, 
HorizontalAlignment.Left, true));

        for (int n = 0; n < 10; n++)
            lv.Items.Add(MakeItem((char)('a' + n)));
    }

    public static void Main()
    {
        Application.Run(new Test());
    }

    private void lv_ColumnClick(object sender, ColumnClickEventArgs e)
    {
        myColumnHeader header = (myColumnHeader)this.lv.Columns[e.Column];

        // toggle the ascending property to sort the other direction
        header.Ascending = !header.Ascending;

        int cItems = this.lv.Items.Count;

        // Turn off display while we load data
        this.lv.BeginUpdate();

        ArrayList rgTemp = new ArrayList();

        for (int i = 0; i < cItems; i++)
            rgTemp.Add(new lviWrapper(this.lv.Items[i], e.Column));

        rgTemp.Sort(0, rgTemp.Count, new 
lviWrapper.lviComparer(header.Ascending));

        this.lv.Items.Clear();

        for (int i = 0; i < cItems; i++)
            this.lv.Items.Add(((lviWrapper)rgTemp[i]).m_Item);

        // Turn display back on
        this.lv.EndUpdate();
    }
}

/// <summary>
///     Class to add a "Ascending" field to a ColumnHeader object so we can
///     determine if the column was last in ascending or descending order.
/// </summary>
public class myColumnHeader : ColumnHeader
{
    public bool Ascending;
    public myColumnHeader(string text, int width,  HorizontalAlignment 
align, bool ascending)
    {
        this.Text = text;
        this.Width = width;
        this.TextAlign = align;
        this.Ascending = ascending;
    }
}


/// <summary>
///     Class to wrap a listview item.  Wrapper class adds sorting 
functionality.
/// </summary>
public class lviWrapper
{
    internal ListViewItem m_Item;
    internal int m_iColumn;

    public lviWrapper (ListViewItem Item, int iColumn)
    {
        m_Item = Item;
        m_iColumn = iColumn;
    }

    public string Text
    {
        get
        {
            return m_Item.SubItems[m_iColumn].Text;
        }
    }

    public class lviComparer : IComparer
    {
        bool ascending;
        public lviComparer(bool ascending)
        {
            this.ascending = ascending;
        }

        public int Compare(object x, object y)
        {
            lviWrapper xlvi = (lviWrapper)x;
            lviWrapper ylvi = (lviWrapper)y;

            string xText = xlvi.m_Item.SubItems[xlvi.m_iColumn].Text;
            string yText = ylvi.m_Item.SubItems[ylvi.m_iColumn].Text;

            return xText.CompareTo(yText) * (this.ascending ? 1 : -1);
        }
    }
}

Thank you Michael Lipp.