What in the primary structure of a protein dictates whether a structure will form alpha-helix or beta-sheet for secondary structure?
The side groups can influence how stable a particular secondary structure might be. For instance, if there are two glutamic acids 4 residues apart in an alpha helix, they would be placed near each other, one over the other, but they would repel each other because of their like electrical charges. In a beta sheet, on the other hand, they would not be near each other. So situations like this influence just what form of secondary structure is preferred for a given region.

Does the formation of secondary structures have to do with the polarity of its side chains? Why do most proteins have extensive regions folded into alpha-helices and beta-pleated sheets?
No. The secondary structures tend to form because of interaction between backbone atoms only. However, side chain interaction could either add to this stabilty or detract from it so as to prevent its formation. For example suppose you had a stretch of polypeptide with glutamates and aspartates at every 4th or 5th position. An alpha helix structure for this stretch would position these negatively charged groups above and below each other, close to one another. One could imagine that the electrical repulsion between these like charges would disallow the formation of such an alpha-helix. One could make an analogous argument about a beta sheet.

In lecture, you've said that "side chain interactions play a major role in allowing or disallowing such secondary structures to form." What sort of side chain reactions prevent beta-pleated sheets from forming?
Interaction between side chain can pull the polypeptide chain out of a secondary structure. Imagine a horizontal beta pleated sheet with a glutamic acid side chain sticking up on one side of the sheet, and now another part of the polypeptide chain oriented above it with an arginine held above the sheet . If the arginine is far enough above it, it could attract the gluatmate carboxyl, thus pulling that residue up out of the sheet. The whole sheet could be weakened by this distortion, and the structure could be resolved by some alternative more stable arrangement, notinvovling secondary structure.

In reference to beta pleated sheets--what is meant by parallel or anti-parallel?
Parallel refers to two polypeptide chains that are both lined up in the same direction, e.g., considered as sections, N terminal on the left and C-terminal on the right for both. Antiparallel is the opposite case, where one strand can be written N to C and its neighbot C to N, and so on for more strands to make a sheet. Beta sheets can form either way, but the antiparallel case is more common, as the polypetide just keeps b doubling back on itself to form the sheet. See the ribbon models of protein 3-D structure in your texts.