private Bitmap bitmap;
private Point oldPosition;

private void Form1_Load(object sender, EventArgs e)
{
    bitmap = new Bitmap(pictureBox1.Width, pictureBox1.Height);
    using (Graphics g = Graphics.FromImage(bitmap))
        g.Clear(Color.White);
    pictureBox1.Image = bitmap;
}

private void pictureBox1_MouseDown(object sender, MouseEventArgs e)
{
    oldPosition = e.Location;
}

* Determine whether the user actually wants to draw (is the left mouse button pressed?).
* Draw the line into the bitmap.
* Update the previous cursor position to the new position.
* Display the results.

private void pictureBox1_MouseMove(object sender, MouseEventArgs e)
{
    if (e.Button == System.Windows.Forms.MouseButtons.Left)
    {
        using (Graphics g = Graphics.FromImage(bitmap))
            g.DrawLine(Pens.Black, oldPosition, e.Location);
        oldPosition = e.Location;
        pictureBox1.Image = bitmap;
    }
}