The strength of an acid is defined by how easily it dissociates. For example, HCl is a strong acid because it readily dissociates in aqueous solution to form H+ and Cl-. In contrast, a weak acid such as CH3COOH only partially dissociates - most remains in the undissociated CH3COOH form.
Why are some acids strong and some acids weak?
There are two main factors influencing acidity - the strength of the H-A bond and the stability of the anion produced upon dissociation. A more stable anion means the acid will dissociate more. This explains why the position of equilibrium of strong acids lies towards the dissociated products. i.e. Cl- is a more stable anion than CH3COO-.
So how does this relate to Ka?
Ka is a measure of dissociation:
This means strong acids - acids that fully dissociate - will have large values of Ka; their position of equilibrium lies over towards the products. In contrast, weak acids have smaller values of Ka since they only partially dissociate. By relating this to the thermodynamic stability of an anion as discussed above, Ka values can become a useful measure of how stable an anion is in solution.
What's the difference between 'concentrated' and 'strong'?
It is a common error to equate 'concentrated' to 'strong', and 'dilute' to 'weak'. However, these are entirely different concepts.
The concentration of an acid depends on how many molecules of an acid there are in a certain volume. The strength of an acid, on the other hand, is an intrinsic thermodynamic property of that acid that relates to its Ka value. It tells you how easily it is able to give up a proton.
It is perfectly possible to have a dilute solution of a strong acid or a concentrated solution of weak acid. These words measure different properties of the acid.