There are many technical responses to this question, but here is the answer most of us can understand. Hooray!
In general, cellular data service is available from your mobile carrier. The service covers a large area - as in miles and miles usually. Customers typically purchase a plan that allows them to download a specific amount of data each period (2 GB, 5 GB, etc.). Cellular data is great for simple internet activities, such as checking email and Facebook, but it tends to be rather slow.
Wi-Fi is an internet signal that is broadcast across smaller areas, such as your home, the airport, your favorite coffee shop, etc. What Wi-Fi lacks in range, it makes up for in speed - it's super fast! Wi-Fi is great for "data intensive" activities such as VoIP (think Skype), streaming video, and downloading music.
Many mobile devices have the ability to connect to one or more of these network signals. Devices with both options often let you choose your connection type. Wi-Fi is a great way to offset what can be costly cellular data plans because, with Wi-Fi, there's usually no limit on the amount of data you are "allowed" to consume. Plus, again, it's faster!
Here's a little tech-talk so you can impress your friends. Cellular data internet connections are broadcast in the 800-900 or 1900-2100 Mhz range. Wi-Fi is broadcast at 2400 Mhz (which means it can not interfere with cellular, in case you were wondering).