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WiFi is the nickname for Wireless Fidelity - high speed internet and network connection without the use of wires, cables and other stuff you can trip over.
WIFI is a wireless network standard certification for wireless networks. Wireless internet is just a slang people use for their wireless home routers. WIFI certification really isn't necessary given that all wireless cards are standards based today. The standards are 802.11a, 802.11b, 802.11g, and 802.11n today.