Weak and strong refer to the degree of dissociation of the hydrogen ion. For example:
HCl --> H+ + Cl-
proceeds near to 100% completion in water. The hydrogen ion and chloride ion dissociate from each other, therefore the term dissociation. This means that if you put 1 mol of HCl into water, you get one mol of hydrogen ions and one mol of chloride ions.
If you have an acid that dissociates to a very high degree (99%+ usually), it is considered a strong acid. If it doesn't, then it's a weak acid. Most acids out there are considered weak acids. When I took HS chemistry, we had to memorize the strong acids. I'll see if I can list them here:
HCl
HBr
HI
HClO4
HBrO4
HIO4
H2SO4
HNO3
HClO3
HBrO3
HIO3
There's probably some that I missed, but that's what I can remember off the top of my head.
Now, for the rest of the acids, they don't fully dissociate. That is to say, if you put 1 mol of acetic acid in water, it doesn't give one mole of hydrogen ions and one mole of acetate ions. The amount that it does dissociate is given by the equilibrium constant, Ka.