Oh, this will be fun.
Basic filters:
HOB (hang-on back) or HOT (hang-on tank) style filters: Very simple to maintain, cheaper than most cannisters, and reliable. A HOB filter is essentially a plastic box that hangs off of the back of the lip of the tank. A water inlet pulls tank water up from the tank, through a set of physical filter floss, a carbon insert, and sometimes a biological filter sponge, and pours it back into the tank. The filter that rykna showed you was a HOB filter. I use a HOB for my H. reidi tank and I've been happy with it. The Emperor 400 style filter is
very nice for a 40-ish gallon tank, with plenty of water movement. It has "biowheels," similar to your 12 gallon eclipse but larger.
Cannister filters: these can either hang on the back of your aquarium or sit below the aquarium. The beauty of these filters is in their water turnover....you can generally find a cannister that moves much more water than a HOB filter. Cannister filters can also go below, so there is less clutter on your tank. The water return on a cannister filter can also be routed into a spraybar so the surface of the water is broken up, increasing gas exchange.
Sumps: First, for a basic seahorse tank, don't worry about 'em. You COULD plumb one in, and in fact it's not a bad idea to consider down the road after you get a feel for the tank. A sump is a separate tank alltogether that sits under your main aquarium....for a 40-ish gallon tank, a sump of 15 to 20 gallons is more than enough. You would use an overflow unit that angs off the back of your tank to route water via natural siphon down to the sump tank. In your sump, you can have all the equipment you'll need to keep the tank healthy: you have a prefilter sponge to filter out the tank gook, then a bag of carbon, then you can fit your protein skimmer and heater in your sump. You can also make a spot for a refugium in the sump, and throw in some chaeto algae and live rock to help your filtration. Finally, you'll need a return pump and fittings to get the tank water back UP to the display tank.
And yes...you could put a cannister or a HOB filter as additional filtration on the back of the sump tank. If you really wanted to.
I love sump filters.....but I really don't think it's something you should try on your first tank. I have a Cascade 300 HOB on the back of my seahorse tank, and it's pretty nice. If I was to put a new tank together with a HOB filter, though, I would probably use the Emperor 400. More flow, and more filtering capability.
Besides which, HOB filters are almost as "plug'n'play" simple as you're going to get. Just slap it on the tank, fill the back with water, add your filter plates and turn it on! Easy-peasy.
I don't have enough personal experience to comment much on cannister filters...if someone else does I hope they chime in. But again, there is a bit of plumbing involved with a cannister filter, and I'm not sure if you want to get into that just yet.
