A hydrogen is the type of bond that links one amino acid to another by the electronegative attraction between an electronegative atom on one amino acid, and the hydrogen bonded to another electronegative atom on a different amino acid. These bonds may be between amino acids quite close or distant to one another in the primary sequence. Hydrogen bonds contribute greatly to the Tertiary Structure of a protein. Hydrogen bonds are about 1/20 of the power of a weak covalent bond, and as such they can be destroyed by forcibly moving elements of the protein. The power of a hydrogen bond lies in the fact that a protein can make so many of them. Anti-parallel beta strands (sheets) will generally form a great number of them, making it difficult to separate them.
As of yet, I am having trouble determining the effect of creating a hydrogen bond on score in Foldit. Generally, any manipulation I make to create one causes a host of deleterious movements within the protein, making isolating the effect of a hydrogen bond formation difficult. Theoretically, hydrogen bonds stabilize a protein, and so should increase score.