The aspect ratio is simply the ratio of one axis to the other: Width/Height.
To retain the aspect ratio, given a target (newWidth, newHeight), you simply have to make sure that (newWidth/newHeight) == (Width/Height)
To achieve this you have to (e.g.) decide whether the width or the height is the most important axis. If you fix width, then you can calculate the height from the width to retain the aspect ratio:
newWidth = newWidth;
newHeight = newWidth * (Height / Width)
To enforce this, you just need to sit on the Resize event so that you control the size whenever any attempt is made to resize your control:
private void Form1_Resize(object sender, System.EventArgs e)
{
sender.Size = new Size(control.Size.Width, control.Size.Width * desiredAspectRatio);
}