The pKa value is the threshold pH (i.e. hydrogen concentration) at which a potential hydrogen acceptor/donator site will accept/donate a hydrogen. To try and make that simpler here's an example: Glycine is a very simple amino acid (H2NCH2COOH) with the structure:

This molecule has two potential hydrogen acceptor/donator sites:
1) the NH2 (amine) site can be further hydrogenated to give NH3+
2) the COOH (carboxyl) site could be dehydrogenated to give COO-
These two sites have different threshold pKa values though. The amine site will become hydrogenated below pH 2.34 and the carboxyl site below 9.60.
Thus as the pH increases (basic to acidic) glycine can be found in three different states:

pH      <2.34       2.34-9.60       >9.60

         NH2           NH3+            NH3+
State    |             |              |
        HCH           HCH            HCH
         |             |              |
         COO-          COO-           COOH
This has obvious effects on it's net charge and further effects on solubility, amongst other things.

The pKa value is the -log10 of the Ka (just as pH is the -log10 of the hydrogen ion concentration). Ka is the association constant that links the concentrations of the free ions and the undissociated parent molecule - the larger the Ka the greater the tendency to dissociate (e.g. very high for molecules that freely dissociate in solution, like NaCl, and very low for things that don't, like sucrose).