A sump / refugium is nothing but a tank that sits under your display tank. Water will flow down into the tank on the bottom from the display tank through either a Hang on the back overflow or a drilled hole in the side of the tank. There is a pump in the lower tank that pushes the water level back up into the display.
You can run the tank without a skimmer as well. I've been running my 20g reef with just powerheads and live rock for about 4 months now. Previous to that it had been running with a refugium and no skimmer for over a year.
HOWEVER things such as a skimmer and a refugium will all contribute to making a more stable system. This will make it easier to keep your corals alive and fish happy.
If you want to start with a low cost system then I would suggest starting out with a basic fish only tank. Don't worry about corals yet. Then save up for the additions such as a refugium or skimmer as you can afford. This will help you get used to dealing with water parameters as well as giving your tank plenty of time to mature.
+1 to that! I'm gonna try and have my cycle started tonight and I don't have lights or a skimmer. It could be another 3 weeks until I get that but the only thing that'll be growing in there is bacteria and they don't need either!
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