A "Mi-Fi" device (the name doesn't really stand for anything) works in much the same way as a 3G USB 'dongle' — it lets a computer connect to the internet using the mobile phone 3G network with broadband-like speeds. The difference is that while a 3G dongle only allows one computer to get online at a time*, a Mi-Fi device can be used with several at once.
This is because rather than connect to a computer's USB port, a Mi-Fi device uses its own Wi-Fi network that other Wi-Fi devices connect to — just like a Wi-Fi broadband router you might have at home. This means that one Mi-Fi device can get any combination of PC and Mac laptops, iPhones, iPads and anything else with Wi-Fi online at the same time — although most devices have a five-device limit.
Mi-Fi devices are battery powered, so can be turned on and left in a coat pocket or bag for use as required. They can also be connected to a computer's USB port and used in the same way as a 3G dongle (which will also recharge the battery), although some models disable their Wi-Fi function in this mode.
Mi-Fi devices are most useful if you have more than computer that needs to get online at the same time using 3G — two laptops in a caravan, for example. They're also a good option if you want to save money with gadgets that you'd have to pay more for to get built-in 3G. The iPad and Kindle both have 3G options, for example, but you can use the cheaper Wi-Fi-only models with a Mi-Fi device.
Mi-Fi devices are more expensive than a 3G USB dongle to buy, but they cost the same to use — depending on which provider and tariff you go with, of course. I use a Mi-Fi device with a T-Mobile PAYG 3G SIM — this costs £2 a day, £7 a week or £15 a month — which is a good option if you only want one for occassional 3G use and don't fancy paying a monthly fee.
Hope this helps — let me know if you need anything else.
(* You can share a 3G USB dongle connection on one Window laptop with one or more other laptops, but a Mi-Fi device makes it considerably easier.)