A sump is basically a space to put things like filtration and other auxiliary items other then your "display" items (What you want people to see in the main tank). It's also a way to add volume to your overall system. It's rare but there are some people that have bigger sumps then they do their main display tank. Basic needs will be things like skimmer, heater and sometimes other filtration like more live rock, filter media, filter socks, chemical dosing or carbon or whatever, a refugium, etc.
You can get away without a sump but you'll need a couple things in your display tank like a heater and a skimmer. Skimmers can be debatable but everyone who gets one will say you need one. It helps to keep the bio loads in check. You can get a hang of the back (HOB) style skimmer for smaller tanks to attach to your main tank if you do not have a sump. Also you can get an HOB filter and use or not use the filter media (also debatable) but it's also great to use if you want to put a bag of carbon or something in there and just to add a little extra flow.
That's how I ran my 29g before I upgraded to a larger tank and a sump. My 29g had a HOB skimmer, HOB filter which I had a bag of carbon in and a heater in the tank. It was ok for a while but I really wanted a better skimmer and more fish so I upgraded to a bigger tank.
By the way, you can run a sump from your main tank in two main ways. You can have an internal over flow (usually the tall black box with cut outs at the top) within the tank it self. Water will go in there and then in a pipe with a whole in the tank and into a sump. It's usually a clean look because you don't have extra exterior boxes and hoses. The other method is to have an HOB external overflow box. This will sit on top of the tank and off the edge of it. Water will spill into a small box on the inside and a syphon tube will bring the water over the top of the aquarium into a box on the outside. From there a hose will take it down to the sump. The HOB external overflow box is a good method if you don't want to drill holes in a tank. Some people get scared of the syphon and if there is a power outage and the syphon still continues. That's a whole other topic. But that's your basic methods of getting water from the display tank to the sump. There's also all in one tanks which have a sump built into the back such has some of the Red Sea Max and Innovative Marine tanks. Then it's one tank with a sump built in and no external mess but they are often more expensive and sometimes less options for auxiliary item setup.
Hope that helps too. Do you have a tank yet? What would you like to do? Just ask any questions you want and people will chime in with some advice. This is a great forum.