Another complicated issue that I will do my best to simplify:
Yes, the laws of Sabbath are based on the Bible, but the Bible is not very specific on this. Based on the Mosaic "Oral" tradition, the "work" that Jews must "rest" from come from the 39 categories of "Work" that were involved in building the Tabernacle in the Sinai desert after leaving Egypt. (Most of the categories are related to farming/agriculture.) From that, multiple subcategories were developed by Rabbinic Jewry over the years. During Temple times there was a Rabbinic "council" called the Sanhedrin, but that was lost after the exile from Israel.
Basically, over time, as new questions arise, Rabbinic Judaism tries to wrestle with these issues based on the original 39 and the subcategories. There is alot of legal analysis here that goes beyond a short post here.
With regard to electricity, it was fairly universally decided that completing an electric circuit would constitute "Boneh," which is one of the 39 and is translated as "building" and is thus forbidden. Contrary to a common misconception, it is not because of the law against starting a fire on the Sabbath, which is another of the 39. Therefore, Jews are forbidden from starting their cars, working on the computer, and... pushing elevator buttons.
I hope I helped. Again, what is forbidden on the Sabbath is an issue that basically requires an understanding of how Jewish Law works, which is extremely complicated.
Just before you say, "The Sabbath is such a restrictive day. What's the point?" There are also Sabbath laws that are not simply "Can't do this" or "Don't do that." Laws such as lighting candles to start the sabbath, blessing over wine, eating festive meals, and the positive commandment to actually rest. The day ends up being a wonderful way for us to slow down the world around us and spend time with family and friends, and dedicate a day to celebrating devotion to G*d. My Sabbath observant family & friends find ourselves frequently saying, "How do people function without the Sabbath?"