Sponge filters are a chunk of sponge or foam that acts as a biological filter. The sponge/foam is porous enough to allow a great deal of beneficial bacteria to set up shop and help keep the aquarium free of ammonia and nitrite. The sponge is attached to a small mechanical thingie-ma-jig (or sometimes just slotted onto a carefully designed plastic thingie-ma-jig).
I'm not 100% certain of how they work exactly but I believe that the air supplied to them by an air pump creates a vaccume that causes particles of dirt/dust/whatever to be sucked into the filter sponge. They are exceptionally simple in their design and absolutely ideal for fry tanks/betta tanks as they have next to no output flow and no inflow that would damage a betta's delicate and generally long finnage.
Pros? I'd say... lack of filter flow, lack of intake suction, adjustable stream of oxygenating bubbles that can prevent stagnantion and bio-film development (the slime on surface of still water), a HUGE house for beneficial bacteria, easy to do maintanence on (just squeeze the sponge out in old tank water once every few months), no mechanical whurring, can be powered using other methods during a power cut (if you have something that can blow air down the tube it'll keep working). They're also great with live plants that don't like filter flow (such as Amazon Frogbit) and won't damage delicate plant leaves.
Kaze liked to watch the bubbles of his stand-alone sponge filter come out and would bite at them on occasion, so they are quite amusing when used with playful bettas. They do still need to be cycled as well.
Cons... lack of filter flow, lack of instake suction (these aren't so good for tanks with messy fish), the air pump can end up exceptionally noisy if it starts vibrating against things (easy to fix though), the filters themselves can be quite unsightly, the sponge can dry out very quickly if taken out of water for any reason and my baby Malaysian Trumpet Snails try to burrow into the foam of one of my filters. Not a bad thing as such as it won't harm them but it's still... random.