Why are weak acids weak?
What are weak acids? 


Examples of weak acids in water solution would be acetic acid (main component of vinegar) and ammonium ion (conjugate acid of ammonia). 


  I understand that an acid is weak if it partial dissociates and breaks down into ionic compounds but they never stated why. 


An acid is weak if not all of the acid molecules ionize into hydrogen protons and its conjugate base in a particular solvent system. Alternately, if we were to use the broader, Brønsted definition, an acid is weak if it does not completely or nearly completely donate its proton to some base. 

Acetic acid is a weak acid in water. Put acetic acid into pure liquid ammonia and now acetic acid will fully dissociate. The proton will be lost from acetic acid to ammonia and this is a favorable reaction. Makes sense, because the only thing that's changed is the solvent that's grabbing the proton - and ammonia is only a few billion times stronger as a base than water. 


  That and do the strong acids dissociate at all. Like if they were in a different solution or do they just stay themselves all the time.


Strong acids by definition ionize to an extent of 100% or nearly 100%. Strong acids will give up their protons completely to the strongest base in the system. 

Note that in the above text I have used the word "ionize" to describe molecular acids because by definition when these acids react, they ionize. Or form ions. On the other hand salts of acids and bases such as sodium hydroxide are said to "dissociate," because ionic compounds are made of ions, and it wouldn't make much sense to say that ions ionize or become ions.