Cliff notes version:
DoF is tied to aperture mostly. Smaller apertures (numbers) allow in more light but give a more shallow DoF. Larger apertures (number) provide greater DoF
So an f/2 image will have a shallower DoF than an f/11 image.
Shutter speed is just how long the shutter is open, and therefore how much light hits the image sensor. When you're shooting with a larger f/stop (aperture). You need to make sure you adjust your shutter speed to account for the smaller overall opening the light comes through. If you don't, your shot may be dark. Keep in mind that if you're down below 1/80 sec or so you may need a tripod to make sure you don't get blurring from your hands shaking. That's a "safe" number, lots of folks can hand-hold down to 1/40 easily. So grain of salt.