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At its core, Wi-Fi is simply this:

Wi-Fi is a certification label ('logo') for wireless data networking products that operate according to the international IEEE 802.11 standard (wireless Ethernet or Wi-Fi). Products that follow this standard use radio frequencies in the 2,4GHz and/or 5,0GHz bands, which may be used without a license under certain conditions. The requirements for this logo are established by the Wi-Fi Alliance.

Source WiKipedia.org

But for most of us, Wi-Fi really means wireless internet. That's what it's all about: we want to connect to the internet to find and share information. And let's be honest, we no longer want to — or can — do without it. So when there's an internet problem, panic sets in, and the Wi-Fi network often gets the blame. 

Final note:

Wireless technology, or radio signals, means working with the laws and principles of physics. We'd love to push those boundaries, but that simply isn't possible. So we'll have to learn to work within the inevitable limitations.

About using the Wi-Fi logo.

The Wi-Fi Alliance does not allow the standalone Wi-Fi logo to be used to indicate that a wireless network is available. To use the official Wi-Fi logo, manufacturers must be members of the Wi-Fi Alliance. There was, however, an exception for the Wi-Fi ZONE logo. Wireless network providers such as WiFi4Rent may use that logo.

If you would like to use a royalty-free Wi-Fi logo, visit the Wi-Fi Alliance website and download a logo of your choice.
http://www.wi-fi.org/who-we-are/our-brands