A "net" is a group of hams getting together in a coordinated, formalized way to do something (e.g. chit-chat or get things done such as coordinate in an emergency or help out during a sporting event).
A repeater is a radio device that listens in on one frequency and retransmits what it hears to another frequency. There's much more complexity to it (e.g. CTSS tones to activate it, it doesn't transmit unless someone's transmitting to it, repeaters are often coordinated to prevent interference), but that's the basic idea. There are two reasons to use a repeater that I can think of offhand. One is that it almost always has more power (being fixed and therefore on AC power, it is limited by laws and what the owner can pay, as opposed to handheld (generally transmits up to 5W) or mobile (often in a car, up to 50W) radios) and therefore can transmit further than a handheld or mobile radio. It also can be positioned such that it can reach much much further than a handheld, mobile, or even stationary radio, and as such can be used to go where other radios can't (they're on VHF or UHF frequencies, which are (almost) line-of-sight), and so they can be used to get the signal out of a valley or over a mountain, or just go further than is possible with one's own radio.