Cat 5 (Category 5 cable) is the type of wire/cable used for Ethernet (100base-T)
It is made from 4 pairs of copper wire (8 all together) that are twisted together. The twisting helps protect the data from interference and noise. Cat 5 cables can be up to 100 meters long
RJ-45 is the type of plug.
Cat 5 is twisted copper cable (not fiber optic). The RJ-45 connectors are used to plug the cable into the computer/router/etc.
Analog telephones use RJ-11 connections which look like smaller versions of RJ-45
Wikipedia has some good pictures of all this:
http://en.wikipedia.org/wiki/Category_5_cable
Related terms:
Cat 5e - used for Gigabit ethernet (1000base-T)
Shielded/Unshielded - most cat 5 cables are unshielded. Shielded cables have a grounded foil wrapper around the twisted wire pairs that protects the data from "noise" (electromagnetic interference)
Full Duplex - This means that Ethernet connections can both send and receive data at the same time (think of a telephone vs. a walkie-talkie. the walkie-talkie is only Half Duplex)
Rx/Tx - Inside of Cat5 cable, two wires are designated to receive data (the Rx wires) and two are designated to send (transmit) data (Tx wires)
Crossover Cable - this type of cable is used to connect two computers/devices without using a router or switch. The crossover cable connects one device's Rx lines with the other device's Tx lines (Basically, the Rx and Tx lines are switched on one of the RJ45 connectors)
Network Interface Card (NIC) - this is the component that allows your computer to connect to the Ethernet network.