If I will be riding a horse in a saddle for a long while or decent amounts of work, I will get it professionally fitted.
But for short amounts of time or not much work (For example I haven't had Latte's saddle fitted yet as I have only been doing half hour rides at the most) then I fit it myself.
First I have a good look at their back - Is it long or short? Is it flat or does it have quite a sway to it? Where is the horses shoulder in relation to their back? Etc.
Then I go have a look at what saddles we have (The luxury of having more saddles than horses :]) and pick out some that I think might work.
To look at fit, I sit the saddle on the back with nothing underneath and my horse relatively clean.
Firt thing I do is make sure it is in the right spot - it needs to clear the back of the scapula to allow the horse freedom of movement. Most kids (Including myself) are taught to put saddle waaay too far forward on the back - I did it for years. It was only when i got a horse who physically told me it hurt, that I realised the error of my ways, and now I preach it to everybody :]
Once the saddle is in the right spot, I look at the size of the saddle comparable to the horses back. A large saddle obvously isn't going to work on a short backed arab - You don't want the saddle to sit too far back over the kidneys. Horses don't have ribs supporting the spine here, so you don't want to much weight too far back.
If the saddle fits the back, then I look at balance. The lowest point of the seat should be the natural place for you to sit, and the pommel and cantle should be fairly even, or as close to even as the saddle design allows (Certain saddles are built with higher pommels or cantles). Basically, it should look even and comfy.
Once the saddle looks good from a distance, I get in close.
First and most obvious thing to check is wither clearance. The saddle should clear the wither by at least a few fingers - Enough that when weight is in the saddle it still has an inch or two of clearance. The saddle should never come into contact with the wither.
Next I walk around to the back and look down the gullet from behind. I check that the pads are clearing the spine all the way through, and that I can see daylight - meaning the channel is free of the spine the whole way through. The pads need to sit an even distance either side of the spine, and not too close to the spine.
Now I will go back to standing beside the saddle, and place a hand on the pommel and one on the cantle, and try to 'rock' the saddle. If the saddle is a good fit, you should only get minimal rocking. If the saddle rocks a lot, it means the pads are too 'bendy' for your horses back - Your horse will need flatter pads. Pads that are too bendy will put more pressure on the center of the back, underneath your seat, instead of spreading it evenly over the entire area.
Next I run my hand under the length of the pads on each side. The pressure should be even all the way along - take care that the sadde doesn't 'bridge' - this is the opposite of the above issue. The pads are too flat for the horses back, creating a 'bridge' effect, where there is a pocket of no/reduced pressure underneath the riders seat. This is one of the worst saddle fit issues and can cause great pain - The saddle will put huge amounts of pressure either side of the withers, and back toward the kidneys, instead of spreading it evenly.
If all the above looks good, I chuck a saddle pad on, girth up and go for a ride. I make sure the horse is feeling comfortable - Most horses will let you know if they don't like the feel. once I'm done, I taek of the saddle and check the sweat patches - they should be even along the back with no disturbed hair. Disturbed hair or dry patches can show spots with too much, or too little pressure.