Sponge filters come in a few different versions, the most common are these two types:
There isn't much difference between these two types, they both work the same way. The air pump forces air through the filter's tubing, which in turn also pulls tank water through the sponge due to the pressure created by the air. The water is then also filtered by the bacteria living on the sponge. This is the safest type of filter you can use for bettas and betta fry because there is no way for them to be sucked into any sort of intake. Sponge filters are also very easy to clean, just squeeze them out in old tank water once a month and they will work indefinitely :)
I recommend an air pump over a powerhead; powerheads have so many different parts to keep straight it's silly. Also, you can easily buy an adjustable air pump with 2-4 outlets, meaning one pump can power 4 tanks, and even more if you get T splitters for your air tubing. I use this type: http://www.ebay.com/itm/Four-Outlet-...item43ac5c7553
Really any kind will work as long as it takes standard width aquarium airline tubing (3/16").
You'll also want to pick up a check valve to put into the tubing if your air pump will be located below your aquarium, to ensure you never get water into the air tube and mess up the pump. Check valves look like this: http://www.ebay.com/itm/10-Non-Retur...item2c6a95732d
Please let me know if you need further clarification on any of this stuff. I know it seems like a lot of stuff compared to a simple intake filter, but it's not too complicated once you understand how it all works, and sponge filters are truly superior and safer.