Strong and weak acids

Acids are proton donors. Bases are proton acceptors.  Acids and bases react by proton transfer.
These are the Bronsted-Lowry definitions. 

H2O can act as both an acid and a base:  H2O(l) + H2O(l)  H3O+ + OH
This reaction is described as strongly reactant-favoured because it occurs to a very small (but measureable) extent in pure water.

Substances referred to as acids are stronger than water. 
Strong acids are stronger than water and stronger than H3O+
Weak acids are stronger than water and weaker than H3O+

The reaction of strong acids with water is product-favoured (in these cases all of the acid dissolved forms products).
HCl(aq) + H2 H3O+ + Cl (aq)
The H3O+ concentration or [H3O+] in the aqueous solution of a strong acid is equal to the concentration of the acid.
Thus the pH can be directly calculated from the  concentration.
For 1
HCl, [H3O+] is also 1
 and the pH is 0. 

Reaction of weak acids with water is reactant-favoured (the extent depending on the acid).
[H3O+] in their aqueous solutions is lower than the initial concentration of the weak acid.
The pH of these solutions is higher than a strong acid in the same concentration.

How do I know if an acid is weak or strong?
There are very few common strong acids.
You probably already know most of them: HCl, HBr, HI, HNO3, H2SO4.  
If you remember that these are strong, it is fair to assume that other acids that you may encounter are weak.