As vk2109 stated, "AirPort" is Apple's marketing term for Wifi (wireless networking).
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AirPort" is used for wireless network connections. You can have a wireless router at home and connect with no wires or you can be at a hotel, the coffee shop, etc., and connect wirelessly.
Apple makes several wireless base stations to make creating a home network easy. See
http://www.apple.com/wifi/
Bluetooth is a wireless technology for connecting devices together over short distances. Typically connections are computer to mouse, keyboard, or cell phone. Also Bluetooth is used between some cell phones and Bluetooth headsets.
Why would I want to set up Airport? Is it meant to have two computers "seeing" each other or just to access the internet? What can it do?
It can do all that.