Open trailer, insert horse, close trailer, drive.
Sorry, I couldn't resist!
Your best bet is to go to a trailer dealer or mechanic and have them give you a demo on whichever trailer you're liking. But I'll give you a run-down on the terminology here, anyway. Hitching up:
Varies by the type of hitch you have.
Bumper pull is where the trailer is hooked onto a ball attached on/near your vehicle's bumper. Usually these are easiest to hitch up to with a spotter to tell you "Go left, now go right, now you're too far right" etc, at least until you're used to it.
Gooseneck is where the ball is located in the bed of a truck. Like a bumper pull, this can be easier if you have a spotter, but I find it easier, since you can sort of see the ball.
Fifth-wheel is not as common, but it's out there. You ever see a semi truck driving around without the trailer attached? That big flat vaguely horse-hoof shaped plate is the receiver for a fifth-wheel hitch; the ball is on the trailer. Like the gooseneck, the plate is in the bed of your truck, and can be even easier to hitch up to, IMO.
(Note that you hear gooseneck and fifth wheel used interchangeably a lot. It's a pet peeve of mine, though, so I always specify.
The trailer receiver will have some sort of mechanism to 'lock' around the ball to keep it from hopping off. This varies by trailer, and I've even seen some that work automatically, but even in that case make doubly sure that it's engaged correctly!
Every trailer ought to have safety chains - big chains attached to the trailer near the receiver, and with hook on the end to snap them to an attachment on either side of the ball. These are for if something goes wrong and the receiver does hop off the ball -- they'll be sort of a last resort hold to keep the trailer from completely detaching, so make sure they're not a rusty, brittle mess!
If your trailer has electric brakes, then there's a little loop of some sort that goes around the ball hitch and, in the event of your trailer detaching, this will yank out of it's port on the trailer and engage the trailer's brakes so it doesn't run away from you. Make sure the cord isn't likely to snap, isn't tangled up in the safety chains, and yank it out of the trailer once or twice to be sure it's not jammed there. This is also handy in making sure your trailer brakes are actually working correctly! Safety Checks
Mechanically, the safest you can be is to make sure you have a mechanic go over it regularly.
As for easy at-home checks.... it's a lot like taking care of your car! Make sure the tires are well inflated, nothing looks like it will fall off, stomp around the floors and be sure there aren't any weak spots (If you have rubber floor mats, make sure to take them out and check the floor underneath too!), regular washing (especially in the horse area -- make sure you at least shovel/hose it out after every use), check the points where you tie your horses to be sure they won't break off, check all over to make sure there's nothing a horse/person could cut itself on. And don't forget to maintain your vehicle!
Every time you go to haul, make doubly sure that the trailer is hitched correctly (see above), all of the lights work, and (if applicable) your brakes work. Brakes
Personally, I see (trailer) brakes as optional for small (like, two-horse) trailers being hauled by a sufficiently large vehicle and driven by someone experienced. If you're not very experienced with hauling, trailer brakes are amazing.... once you learn how to adjust them properly. And that is something that varies so much by trailer and by controller (which is the piece in your vehicle that tells the trailer brakes how much force to apply) that you'd be best off asking that previously mentioned mechanic. Types of Horse trailers
Your most basic type is the stock trailer. These have a big open space, sometimes with a swing gate about halfway back to divide it into sections. There are generally open slats that run the length of the trailer near the top of the walls to serve as windows -- some have plexiglass/etc over them, some are just open. These trailer offer a lot more room for claustrophobic horses, and they can comfortably stand in whichever direction they like. Some people tie, some just leave them free roaming in the trailer.
Slant loads are quite common. The horses are loaded standing at an angle, and depending on the exact makeup of the trailer, the horse might be required to walk into the stall without a person at it's head, and then back out of the trailer on command. Most slant loads have drop down windows that the horse can stick it's head out of (but never leave these windows open while you drive down the road!), some with bars or nets that allow you to leave the window open but prevent the horse from hanging it's head out (you can usually get these aftermarket, too). Most also have some sort of tack room -- a small triangular room is common at the back of the trailer, as well as a somewhat larger one at the front.
There are also a few varieties (like my own) that have removable dividers for the stalls and a collapsible rear tack, so you can convert it into a stock trailer.
Straight loads are common as two-horse trailers, though they certainly come larger. They'll usually have a hay manger up front, and the horse must walk straight in, and then back out when you're ready to unload.
You can also get a trailer with a ramp or a step up. Ramps are nice for horses that are clumsy, and resistant to loading, but the ramps can also be cumbersome to raise and lower, plus you have to make sure there's room to set it up.
There are varying opinions on how a horse 'prefers' to stand in a trailer, and what's most comfortable for them, but I'm of the opinion that it differs by horse -- one of my mares will stand facing front every chance she has, while my two older mares always stand at a slant or sideways, and my geldings like to face the back of the trailer!
There's also a matter of how big your horse is. Very tall horses can feel cramped or hit their heads in a "standard" height trailer, so look for a taller one. Large drafts likely aren't going to fit in the stalls of your usual straight or slant load. Small ponies might be able to slip underneath the dividers in some straight or slant loads. For this reason, I have a slant with a high ceiling and the dividers removed, to safely and comfortably fit my horses that range from 11-16hh.
As for the matter of whether or not to tie your horse or not... it largely depends on preference, your horse, and your type of trailer. I don't suggest leaving horses loose in a stock-type, because they might tend to wander and could end up standing at a spot that would cause the weight to be unevenly distributed, making the trailer haul awkwardly.
If you tie them up, make sure they have enough slack that, should they fall, they wouldn't be dangling by their halter and would have a chance of getting up again; but don't leave so much loose rope that they could wrap it around something. It's also handy to make sure you would be able to untie them from outside the trailer if need be.
It's likely I forgot something, so maybe someone else can point that out, or you can feel free to ask~