The simple concept.
Water falls from the tank by gravity into the sump. Either with an overflow box, or by drilling tank.
I prefer drilling because it is less likely to fail, in in the long term, less expensive, I have drilled at least 8 tanks with the same bit.
Entering the sump water passes thru mechanical media. I like filter socks, easy to pull and rinse
In the sump is biological media. Can be anything from ceramic rings, sponges, Scubbies, to lava rock.
I like to put bio-media in mesh bags, easy to remove and rinse
The sump can also be a Refugium containing plants and lower animals (Caulerpa algae is great for salt water) that help process waste, and consume nitrate. Below is a 40 gallon planted sump/Refugium, during a water change.
Below same sump full, pic taken from the side
For salt water a protein skimmer is a given. DIY models are everywhere on the net.
Below is a video of the bio-fractionator I use on fresh water tanks and ponds.
koi pond fractionation
Fractionation helps lessen the need for water changes, one of the most on-going expenses of a salt tank, compared to fresh.
Sumps can also contain heaters, and other unsightly equipment.
And of course will hold the pump that sends water back to the tank.
One of the most important factors of a sumps, is placing it, and sizing it so maintenance is easy to do.
I often place them off to the side of a tank instead of directly under, to make it easy.
And when I use 50 gallon barrels for sumps the don't fit under allowing room for arms anyway.
In the photo above I have a DIY fluidized bed reactor to the left of the barrel sump, made of clear PVC.