If you are interested in the brightly colored cichlids, I'm guessing you would prefer mbuna (means rock-dwelling), one of 3 groups of cichlids from Lake Malawi, Africa.
There is a lot of research needed to successfully keep them, they aren't difficult in terms of feeding or water quality, but rather grouping species that are unlikely to destroy each other.

They are quite aggressive and temperaments can range from mildly aggressive to unbelievably psychotic. And because of their temperament and preference of hard alkaline water, mbuna should only be kept with other mbuna with the exception of some catfish.
For the basic requirements:
Tank: Most mbuna species can be kept in a minimum 4ft tank, they are mostly bottom dwellers and while they generally average around 5 inches or less, they are very territorial and need plenty of space. A couple of species get a bit larger than average, notably
Pseudotropheus acei & Pseudotropheus crabro which both reach around 7-8 inches and should really be kept in a 6ft tank.
Filtration: As a couple of people have said you will need higher than average filtration. Mbuna should be overstocked to cut down on aggression, this allows sub-dominate or weaker fish to get "lost in the crowd" thus confusing the dominate fish (usually a male) who will randomly chase anyone near by.

So, as a minimum you'll need a total tank turn over of 8-10 times. This can be achieved with a very large or multiple filters (should be the bulk of your turn over) and pumps/powerheads. You can go much higher with your turn over as well as long as none of the fish are getting blown around the tank. For example, I'll be setting up a new tank with a total turn over of around 25-30 times per hour!
Substrate: Mbuna thrive in high PH (7.8+) and hard water, so substrates that will buffer your water are preferred, that is unless your tap water can be considered liquid rock.

Mbuna love sand, they'll happily dig in it all day long and it's great fun to watch; for their buffering ability, aragonite or coral sand are tops.

You can find either in your LFS marine section, it does cost a bit more than play sand, but it's worth it.
Decor: As I said above, mbuna means rock-dwelling and that's exactly what they are. In the wild they live, eat and breed in rock piles, so your tank will need lots of rocks or other decor that will allow them to stake out territories. To help cut down on fights try to have multiple caves/crevices for each fish. For some inspiration check here:
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Compatibility: Here's where it gets tricky, it's always important to research any fish before you buy it, but it's doubly so with mbuna. The biggest problem new keepers have is putting together incompatible species in the wrong ratios. The vast majority of species should be kept in ratios of 1 male for every 3 females. There are some exceptions of course,
Labidochromis caeruleus & Pseudotropheus acei are both mild enough that they can generally be kept with multiple males. On the other hand, those species with higher aggression levels should be kept with only 1 male per tank and 4-5 females (per species). And then there are some species that shouldn't be sold at all as they are notorious for killing their tankmates regardless of how carefully you stock it.
Melanochromis auratus & Melanochromis chipokae should be avoided at all costs, in fact I don't recommend any of the
Melanochromis species for beginner mbuna keepers.
Metriaclima lombardoi, commonly called Kennyi, are also very aggressive though they can be kept in larger tanks with other aggressive species. And then there's
Pseudotropheus demasoni, one of my personal favorites, they are the biggest exception to the 1m/3f ratio. They have such high conspecific aggression (aggressive towards their own species and those similar looking) that they should be kept singly or in groups of 12 or more. That said, they do ignore any species that doesn't look similar so they're still a good choice.
Have I overwhelmed you yet?
You will notice they I'm using scientific names, this is really important with mbuna, there are no real common names for them like say a neon tetra, each LFS can and will call them whatever they want and so when identifying them you should always use their proper name. Another thing that should be avoided is picking any fish from a Mixed African Tank, these often contain hybrids which should be avoided for two reasons. 1 there are so many hybrids on the market now that it is becoming difficult to find pure strains and 2 you don't know who the parents are, the cute little fishy you pick out could turn out to be a killing machine.
I generally recommend to new keepers to go to their LFS and write down all the names of the fish they like the look of, then research them
here or
here and then post on here for help with compatibility. Grouping mbuna is just as much an art as it is a science, research and experience will tell you what will work and what won't which is why asking other keepers is so important. There are a couple of guidelines to follow though, avoid keeping species that all look alike and try to keep those with similar temperaments.
Further Reading: I highly recommend reading the following, as I said research is key to a successful mbuna tank or any tank for that matter.
The Mbuna Aquarium
Feeding Mbuna
African Cichlids, multiple articles
Well, if you've gotten through all that and still have questions, feel free to ask and I'll do my best to answer.

-Dawn