PKA

The pK_a 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:

    NH2
    |
   HCH
    |
    COOH

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 pK_a 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 pK_a value is the -log₁₀ of the K_a (just as pH is the -log₁₀ of the hydrogen ion concentration). K_a is the association constant that links the concentrations of the free ions and the undissociated parent molecule - the larger the K_a 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).