each computer has to have it's own unique IP address on the local network. but as default all computers are set to obtain an IP address automatically and will query the router to obtain one when you connect them either by wire or wireless.
Though they are private IP addresses and are only used on your local network (the computers connected to your router)
you also have a public IP address, which is your internet address, the router has this address and it job i to "route" all traffic to the correct computer on your local network using it's unique IP address .
http://en.wikipedia.org/wiki/Network_address_t ranslation
for a better explanation.
The average user doesn't need to worry about IP addresses though.... it's all done automatically (and it works most the time!)