Wi-Fi = a wireless connection to your home Internet and/or Internet provided by businesses such as McDonald's, Starbucks, etc. The same as having your home computer/laptop/smartphone hooked up to your modem but instead of using wires you're using a wireless connection. The range provided by a wireless router tends to be very minimal.
3G = a "cellular" wireless connection where you connect to the Internet via a cellular service provider's signal (AT&T, Sprint, etc). Anywhere you can get a signal on your phone you can access the Internet via 3G. To use 3G internet you must either have a mobile phone with a data plan or a 3G access card (USB stick) provided by a cellular company for your computer/laptop.
Wi-Fi connections tend to be quite a bit faster than 3G connections but there are variables such as the base speed of the Internet connection, the wireless technology being used (B/G/N), and how far away you are from the transmitting device. Most Wi-Fi connections are free except for those generally found in hotels and airports. If you have a Wi-Fi connection at home then of course it will be free.