Since you seem to be trying to be provocative on your other thread to get an answer to this I'll try although I'm far from an expert. There are many different types of suspension forks and you should get one that matches your bike and your style of riding. Given the many different companies that produce forks you will either have to do your own research or ask about a specific fork model.
There are a few important ways in which forks differ from one another. First, travel refers to the usable amount of suspension on a bike and is measured in millimeters. For example, an 80 millimeter travel bike has eight centimeters of usable travel. The amount of travel you need is governed by your riding style and your bike. Typically XC riding (riding on smooth or relatively non-aggressive trails) uses between 80 and 100 millimeters while all mountain uses up to 140 depending on the bike. Be aware that bikes are designed for forks with a certain amounts of travel and that if you mount a fork with too much travel it may stress the frame to the point that it can fail. Generally, there is some play in how much travel a particular frame can take but you should check and see what comes stock on your bike to get an idea of a starting point.
Second, forks differ in the types of suspension they use. There are two systems currently being used, coil and air. Coils are essentially springs in the fork which can be tensioned to give the ride you want. The advantages of coils is that the travel on some model forks can be adjusted on the fly without the need of any special equipment. Plus some riders just like them. Air suspension uses pneumatic pressure in multiple chambers of the fork legs. Typically these systems are more expensive but they provide a smoother more "plush" ride. To set up an air fork you need a fork pump (think tire pump but for forks) so it's something you have to set for your ride and forget. I believe (please correct me if I'm wrong) that many forks have the suspension device in one leg.
The third factor is the ability to adjust the forks. Most decent forks come with at least preload. Preloads purpose is to set the sag of the fork when you sit on it. You want the fork to sag about an inch when you are doing nothing but sitting on the bike. This is because the primary purpose of the fork is not necessarily to provide you with the smoothest possible ride (although that is nice) but to keep the wheel firmly pressed against the ground as you travel over uneven terrain. Next, many forks have a lock out option. Lock out keeps the fork from bobbing up and down and is used so that riders do not waste energy when pedaling up steep non-technical fireroads and terrain. Often times you can get handlebar mounted lockout devices (such as poploc). As noted earlier coil forks may have travel adjustments.... With air suspension forks the number of chambers is a key point for setup. I'm less familiar with these systems but they have a positive and negative chamber that can differ in air pressure. This difference will effect the quality of ride. If you're serious about getting one of these you should ask someone that knows more about airsprung forks...but if you're just getting into mountain biking it's probably unnecessary.
On a final note, don't over think the fork. I ride a relatively cheap hardtail with a budget fork and have beaten the socks of guys riding bikes 5 times more expensive than mine with nice forks. Back in the day they used to ride rigid and it's coming back especially with the 29er large wheel bikes that are able to soak up more impact in the wheel and tire. Reputable fork makers include: Rockshox, Manitou, Marzocchi, Fox and Magura. Good luck.
Some of the stuff I look for:
Seals not leaking (compress the fork, see if they leave a ring around the stanchions)
No headset play
No dents on frame (scratches I'll deal with, dents not so much)
No rust (indicates bike may have lived outside and/or wasn't taken care of well)
Shifts fairly smoothly (and if not, see if it's just a matter of replacing the cables or if there's a bigger problem)
Drivetrain isn't beat to death with all its teeth worn to nothing
Cheap non-stock parts thrown on (suggests components broke and were replaced with cheap stuff)
Anything else that suggests a lot of abuse. Yeah, bikes are meant to be abused, but if I'm buying myself a used one, I'd prefer one that hasn't been abused to all hell and may have some other problems yet to be discovered with it.
Also, these problems can be fixed with replacement parts, but unless it's get a screaming deal on a really nice bike, it's sometimes more expensive to fix up the bike than the bike is worth, especially if it's an old or inexpensive bike to begin with. There's something to be said for the challenge, but some of the fixes can get really pricey and end up being almost as expensive as a new bike.
Yeah, you do need to check bike fit and test ride the bike. Figured that was a given so I didn't list it. And you shouldn't need the front tire inflated (or a front wheel at all) to check whether the seals are blown. Just set the fork down on the dropouts (or on the front rim) and compress.
It seems like you've bought you bike. For future reference the travel should be smooth. There should be no rust on the stanchions and no visible damage to the fork legs. Check any mechanisms for adjustment and see that they work. You should never be able to easily bottom out a fork.