It sounds like you've got a fairly broad range of trailers that will work for you. The harbor freight trailer linked above will likely work for a broad range of lightweight 12' boats. It's a "bunk" trailer made of what looks to be painted or powder coated steel.
In trailer terms "bunks" are the carpet covered wood that your boat will sit on. Bunks can also be made out of bare wood or plastic wood type material. Other trailers might have rollers (rubber cylinders the boat will roll over), or a combination of bunks and rollers. The type of boat you have dictates if you should have bunks, rollers or both. Generally small flat bottom or nearly flat bottom boats (such as the classic 13' Boston Whaler cathedral or tri hull style) use bunks while traditional v hull boats will use rollers under the keel either with more rollers or bunks along the sides of the v.
Steel, either painted, powder coated, or galvanized and aluminum are pretty much the only materials trailers are made of. Painted or powder coated steel is the cheapest but it's the least durable. It's likely OK for freshwater but you need to keep a close eye on it for rust. Galvanized steel is a better choice, especially for saltwater applications but it will eventually rust too, especially where holes are drilled to attach bunks, rollers, suspension, etc. Aluminum makes for a corrosion resistant lightweight trailer but it's expensive and can still have problems too, particularly where dissimilar metals meet (such as steel bolts and axles).
Almost all the trailers you'll be looking at will have leaf spring suspension and steel axles. Even in galvanized steel or aluminum trailers you'll find plain steel axles and suspension members. These are the weak points for rust, and trailer owners often forget to check them after the boat has been sitting on the trailer in the off season. You may also encounter torsion axles (the spindle is offset from the main axle by a lever that twists rather than riding on leaf springs) but they are rare for boat trailers. You also won't need trailer brakes so I won't get into those.
The axles, bearings, wheels and tires are going to be the most difficult and important part of a trailer to maintain (well electrical is a pain but can be easily solved, which I'll get to later). Inspect your wheels every trip, check for water coming out of the axle hub after every time you launch or haul your boat and check for slop/play in the bearings. Keep them properly greased (but don't blow out the seal by over greasing). Rather than keeping just a spare tire mounted on a rim I suggest you keep a spare tire, rim, hub and bearings all greased up and wrapped in a plastic bag. If something goes wrong with a wheel it's as likely to be bearing and hub related as it is to be rim and tire related. Of course it's also likely the axle spindle or suspension break but you can be prepared to solve tire, rim, hub and bearing problems on the roadside easily.
For small boat trailers I recommend you rip out the electrical system entirely and just use a light bar. That's a 2x4 slightly wider than your boat with the trailer lights (and license plate) mounted on it, and ropes at each end. The ropes go down from the light bar, around the frame of the trailer and back up to cleats on the boat or the light bar itself. Then a length of wire runs freely from the light bar tied at the stern of your boat up to the bow and to the plug on your tow vehicle. Secure the wire near the bow so it doesn't scrape on the ground and you've now got a light system that doesn't get submerged and can be fixed without having to slide under the trailer. Remove the light bar as you prepare to launch and tie it in place once you haul out. It even secures the boat to the trailer.
When you are towing you want the trailer to sit level, with about 10% (oversimplification but good starting point) of the weight at the tongue of the trailer. In other words if the whole boat, trailer and gear weighs 1,500 pounds then a scale right where the hitch goes should read 150 lbs. Don't forget to take into account the engine, gas, battery and gear you load into the boat and how you load it in relation to the trailer axle.
There's plenty more to know but that should get you started.
