You are correct, a smaller number means a larger opening (apeture) which means a smaller depth of field (or focus)
The larger apeture also means you get more light in, so you can have a faster shutter speed.
Apetures range from f1 (very very very fast) up to f91on my camera (very very very small)
In reality unless you have a really expensive lens you may only see values between f4 and f27.
The values of the shutter speed and apeture are linked, say you had the correct exposure for a scene and you then changed the apeture to a higher number, to get all the scene in focus, to get to correct exposure again you would have to let more light in, because yopu have made the apeture (hole) smaller. You would do this by keeping the shutter open for longer (a slower speed).
The values of both are measured in stops, and each stop lets in twice, or half, the light in. This is easy in shutter speed, because if the shutter speed is double (1/50th second to 1/25th second) then it will let twice as much light in.
In the apeture it is more difficult because the numbers don't really mean that much, here are the apeture f-stops:
f1.8 f2.8 f4 f5.6 f8 f11 f16 f22 and so on.
So the following combination of shutter speed and apeture will give you EXACTLY the same exposure, but will have different depths-of-fields.
F1.8 at 1/250th second
F2.8 at 1/125th second
F4 at 1/60th second
F5.6 at 1/30th second
F8 at 1/15th second
F11 at 1/8th second
F16 at 1/4 second
I hope I have explained this right, it has been a while since I learnt it all.
This might also help:
http://www.photoxels.com/tutorial_aperture.html