I'm replacing my shower and I will have to move the drain because the new shower has a drain in a VERY different spot. Is is possible to elbow the drain line and connect a pipe that will reach the new shower's drain? It seems like drain lines are always straight, but then again I know VERY little about plumbing.

I think you are asking about moving Trap riser only. ( Trap riser is the pipe that sticks straight out the ground and connects to drain head ). That section of trap should be straight and w/o any bends.

I get impression from your question that new drain is not in proximity of the old drain. Therefore, you would probably have to use 90 degree ells to move it over. That would be too much and water would not drain properly. I have seen 2-45 degree ells in trap raiser before - but that was in situation where drain was moved about 2" and trap riser was long enough to absorb the offsetting.

Now if you want to move your horizontal drain bellow the grade, angle it towards new location, and reinstalling trap - then I would suggest that you use 45 degree bends instead of one 90 ell. . If you don't have space and you have to use sharp bend - then use 2-street 45 degree bends.