Shabbat is the weekly celebration of the Creation story in Genesis. In the story, G-d creates the world in six days, and on the seventh He rests. To remind us of this (as we are commanded to do in the Ten Commandments), we rest on the same day.
Shabbat comes in at sundown on the Friday night, and goes out at the Havdalah service on Saturday night.
Jews go to shul on Friday night, and then again on Saturday morning, where we ready the week's Sedra (portion from the Torah).