UGF's (undergravel filters) do not do anything as far as filtering goes.
For a decent length of time they were pretty much the only filtration option available and worked fine- for 20 years I never had a problem using one.
@ animation:
Alright? I'm going to give you top to bottom on filters, so I don't miss anything. Apologies if it sounds patronising!
In answer to your question they draw water down through the gravel (it uses a tray to suck down as uniformly as possible across the base of the tank), and up a tube back into the tank (the pipe screws into the tray). You put the UGF in before adding substrate. The substrate is then effectively used as filter media. How you draw the water down through the gravel and up the pipe is the question.
In theory you can dangle a little air stone (connected to an air pump) in the middle of the pipe, and the upwards motion of the bubbles pulls water with it; which then gets replaced by water drawn through the gravel etc. etc. Personally I question the rate of flow when using this solution. Perhaps Durbkat took this route, I don't know.
The second method is by using a power head. These are basically pumps that sit on top of the pipe (watertight seal is required) and draw water in from the pipe (and therefore down through the gravel). These pumps can chuck out really decent amounts of water, and do keep a proper flow going through the substrate.
UGFs are not suitable for use with sand substrate as (a) it compacts and water can't flow down through it, and (B) it can get easily sucked up into your powerhead and knacker it for want of a better expression.
Also as was mentioned they aren't much good for plants either as they suck water over the roots too fast to allow nutrient absorbtion, put simply. They do suck loads of debris down through the gravel, and it does accumulate. If you get the opportunity to look up at the bottom of a tank using UGF after a few months, you wouldn't believe the amount of black decomposing matter you will see.
There are alternative filtration methods available, including canister filters, HOB filters, external filters, and sumps.
Canister filters are basically a powerhead with a cube of foam enclosed in a plastic casing stuck to the bottom. The foam makes an ideal place for nitrifying bacteria to live (the bacteria that changes toxic ammonia- fish waste- to less toxic nitrates, and also the different bacteria that changes nitrates to even less toxic nitrites), instead of using your gravel. You can unclip the plastic casing and rinse all the rubbish out of your filter (in old tank water since chlorine in tap water kills the bacteria), hence people saying they are easier to maintain. Canister filtetrs are available in different sizes according to tank volume.
HOB filters are very similar indeed, except they Hang On (the) Back- HOB. They have a pipe going into the water to suck it out, and another going back into the tank. This stops you wasting tank space with a canister filter.
External filters are the same again, but for bigger tanks. They are too big to HOB so sit on the floor with pipes running to and from the tank. These contain lots of foam/other filter media, as well as having more powerful pumps so can deal with more waste.
Final step up, for massive tanks are sumps. These are generally the reserve of saltwater fishkeepers. The idea is the same, but you want that much media that it is easier to use another tank sat underneath (and hidden). Pumps send the water down, send it through often multiple stages of filtration, then pump it back up into the main tank. Besides the advantage of being able to have loads of media to filter lots of waste, there is one more advantage- increased water volume of the whole system. Often a sump is 1/4 of the size of the main tank. This means that instead of for example you having 200 gallons to dilute waste or toxins in, you have 250 gallons. Toxins and pollution are therefore less concentrated and less harmful to your fish.
The final extension of this you might see in a fish shop. Some link all their tanks into a massive sump many times the volume of any one tank. This allows them to monitor the quality of a single huge volume of water- but it does introduce the potential for a single sick fish to introduce disease to the whole system!
Have a think about your options, and what equipment you might need to buy. If you have to spend £20 on an air pump, might you not as well spend it on a canister filter? Up to you, ask if you have any further questions.
As a footnote, many people keep Bettas in unfiltered tanks (but with regular water changes). They live in the wild in crappy water, and are more resiliant to pollution than most. I wouldn't condone this particularly, but safe to say you're not going to need a sump or massive external filter. Your UGF or a small internal will do the job nicely.