Slip angle is just the angle between the direction the tire is pointing versus travelling. I.e., you can project the tire side vector (the axis the tire spins around) onto the ground, then do a dot product with that and the velocity vector of the center of the tire to get slip angle. Aligning torque doesn't "cause" slip angle at all, rather it's the other way around. Slip angle is just the angle.
Aligning torque is caused by slip angle, essentially. In that Pacejka paper earlier in the thread there are some diagrams that show what's happening in the contact patch in the case of pure slip (where slip angle exists, but there's no slip ratio). Rubber enters the contact patch a little bit pulled to the side of the wheel center plane, then is progressively pulled further and further to the side as it travels toward the rear. At some point it then slips back toward the center.
What happens is that the "force centroid," or the effective center of that force, is not in the center of the tire in most cases. The force centroid is like a center of gravity really. All the little forces throughout the contact patch, when added up, are the same as some bigger force acting at one specific spot in the patch. You can see more distortion in the rear of the tire than the front in most cases, so it should be fairly intuitive that the force centroid is usually toward the rear of the tire.
This distance from the center of the tire to the force centroid is called the "pneumatic trail," and since that force is acting behind the center of the tire it tries to twist the tire. Generally it tries to straighten it up, but actually at really large slip angles it can reverse itself (i.e., it makes a torque). The aligning torque is simply this pneumatic trail times the lateral force.
Caster angle then adds an additional distance to that pneumatic trail. I.e., a line through the steering axis intersects the ground at a point. The lateral force acts behind that, so you get an additional torque. The distance from that point to the center of the tire is called the "mechanical trail." So aligning torque becomes (mechanical trail + pneumatic trail) * lateral force.
The aligning torque really doesn't do much to the handling. All it really indicates is that the force centroid is moving a little bit forward/rearward as a function of slip angle. This primarily is just a steering force feedback thing happening.
What about slip ratio and aligning torque?
The simpler answer is that slip ratio produces a forward force in the tire plane. If that force acts in the center of the contact patch along the width of the tire (even if it's toward the rear or front of center), then there would not be any aligning torque contribution due to slip ratio. The force goes straight forward in the tire plane so there's no torque generated.
In reality the situation is not quite so perfect. The force centroid could very well be slightly to the left/right of center (especially with camber) which indeed would give you a little different aligning torque once you change the slip ratio via throttle/brakes. The effect is generally not very big though, and again, for the most part all this is doing is changing the feedback through the steering wheel. Moving the center of force at the tire around in the contact patch a couple of inches isn't going to make a noticable difference in the handling. Although the feel through the steering wheel would be different, and some people appear to judge the handling by the force feedback more than they do the actual balance/understeer/oversteer characteristics of the car.
A little on Pacejka: First, Dr. Pacejka has written a TON of tire models, not just the well known Magic Formula used in Racer and several other sims. I've got a book here published in 1971 that has several other tire models written by Dr. Pacejka, and they're quite advanced. He's been doing tire models for at least 35 years then in that case. There's a reason his Magic Tire model is used in real vehicle dynamics research. It's possible to reproduce real tire test data very accurately with it.
The snag that sim developers run into when trying to implement specifically his Magic Formula is in combining lateral/longitudinal forces. I.e., if you use that formula with say 4 degrees slip angle and no slip ratio, and the constants are chosen correctly to match up with a real tire, the force and aligning moment you get out of the model matches the real tire exactly. If you ran 0 slip angle and 0.05 slip ratio in the same equation with the proper constants for longitudinal slip, again, you'll get the right force.
However, once you start putting in slip angle and slip ratio at the same time, the forces all change. I.e., 4 degrees slip angle with 0 slip ratio produces a different lateral force than 4 deg and 0.05 slip ratio does. If that's not done correctly it impacts the handling in a major way. I think LFS had this problem all along until the latest update, where this now works much better. The result is a car that's easier to drive for sure, which is realistic. Don't let yourselves be fooled into hard=realistic. It just isn't the case at all...
If you get a chance to play the old arcade game Hard Drivin' or Race Drivin', give it a go. That's a vehicle model created by Doug Milliken and associates of a Corvette that uses a tire model with real tire data. I.e., that's a professional engineering model you're driving there that's dead accurate. Is it hard to drive? Not at all compared to a lot of sims, but it sure was harder to drive than Pole Position or any of the arcade games that were out at the time it was released.
You could also try Silicon Motor Speedway, the Nascar simulation. This was done by Doug Milliken too (in fact, they have a Milliken Raceway in his honor that the employees run on occassion). That again uses real tire data and is the same model they use for real research and engineering on real cars.
So if you guys are looking for a benchmark for simulation reality, try those titles and then judge all these other sims based on those.
, merely tyre compound wouldn't be all that accurate) and as such the force with which that bit of tyre tries to move back to its original place both laterally and longitudinally... then you'd probably "multiply" that by the "contact patch load distribution diagram" to get the actual force on the road that that bit of tyre produces. Do that for the whole tyre and you could end up with the force centeroid position, and a lateral and longitudinal force... I know this is rather ambitious for real time processing but you could put this all in a table and attempt to generate equations from there or alternatively make a "canned" engine. (I know I've expressed my negative thoughts on those but maybe... if you got an accurate enough lookup table... just maybe it would work). Or do I have something inherently wrong with my theory? I can see it generating simmilar curves to what they are in reality. If you don't distort the contact patch enough you don't have enough displacement to produce a sufficient force. If you distort it too much your contact patch distribution is messy.
Todd? 
Runs in real time with no problem at all; see Virtual RC Racing. There are no look up tables or anything like that in there either.