I'm referring to the score in the bottom right corner of the screen. I was running out of turns and was behind in points so I don't know what I should be doing to make it better.

Does the score correspond to a year, like this is how advanced this civ is?

I don't know the formula or anything, but it seems to be based mainly on territory and techs. Also, it looks like gold and units play a slightly lesser role in the score.

I believe if you hover your mouse over your name/score at the bottom right of the screen (in the area with all the other Civ's leader's names that you can click on to instigate diplomacy) it gives you a breakdown of how your score if being tallied. Hovering your mouse over your rival's names does nothing, though.

Yeah mousing over gives a breakdown. It seems to be based on population, territory, technology, wonders and future tech. Maybe there are more ways to get score and I just haven't done them in this game, though.