Basic definition of LBWA batsman is adjudicated LBW, when the ball hits his pads and the umpire thinks that the delivery would have gone on to hit the stumps.
Leg Before Wicket, or as most of us call LBW, is in a way to cricket what the offside rule is to football―most people think they know it, but many don't understand it completely or can't explain it properly. I have met many cricket fans who do not understand the exact rules for judging an LBW dismissal, and trust me, there is more to it than meets the eye. It is very common to hear shouts like "that was plumb!", or "why the hell didn't he give that out", and so on, while watching a cricket match. Many people intuitively decide whether or not a batsman is out, or whether or not the umpire's call is the right one. Now, with slow motion replays and the other visual aids, a television viewer has a lot more at his disposal to decide, provided he knows the basic rules of LBW.
Of course, if it were that simple, there would have been no confusion at all about this mode of dismissal. But, there are some exceptions, and also a few preconditions.
First, let us clarify how the umpire decides whether the ball would have gone on to hit the stumps or not. There are two things to consider: the line of the ball, and the height of the ball when it hit the batsman's pads. Umpires extend the perceived line in their mind (calculating any lateral swing/seam movement), till the stumps. They also consider the height at which it hit the pads, and decide whether the ball would have gone over the stumps or not. There is an unwritten rule: if there is any doubt in the umpire's mind, the benefit of doubt must go to the batsmen (which holds true in general for cricket). So, if the ball is spinning/swinging too much, it would be very unlikely for the umpires to give the batsmen out LBW, unless the batsmen is way back in his crease.
Provided that they think it would have hit the stumps, there are other preconditions that umpires need to take into consideration. A batsman is judged 'not out' if:
a) There was a prior contact of the ball with the bat.
b) The ball was pitched outside the line of the leg stump.
c) The ball has hit the batsman outside the line of the off stump.
Now, let us consider these factors one by one. If there is any edge prior to the ball hitting the pads, the batsmen cannot be given out LBW. Remember that the edge has to be before the ball hits the pad, and not after. In the latter case, the edge does not matter. If the ball is pitched outside the line of the leg stump, there is just no way that a batsman can be given out LBW. Some people are puzzled by this rule, but I think it is very logical. Consider the fact that, when the ball is pitched outside the line of the leg stump, the pads naturally come first, obstructing the shot at times, whereas, in other cases, the ball hits the pad only when it misses the bat. So, there is a need to exclude cases where the ball is pitched outside the leg stump. When the ball hits the batsmen outside the line of the off stump, the batsmen cannot be given out LBW either, provided he was attempting to play a shot (or, in cricketing terminology, offering a shot). This is also confusing for some, but again, very logical. First of all, when the ball is pitched outside the off stump, the batsman may have to take a stride to reach the ball, which means, his pad might again come in the way of the shot. However, it's deliberate padding, i.e., not offering a shot, and just taking a stride to take the ball on the pads, that's unfair. So, a clear distinction is made there, between the batsman offering and not offering a shot. In the former case, the batsmen is given 'not out', whereas, in the latter, he is given 'out', even if the ball hits him outside the line of the off stump.
This covers the LBW dismissal's complexities. To recap, a batsman is out LBW if the ball hits the pad with:
1) No prior contact with the bat.
2) Ball pitched in line with the stumps OR outside the off stump.
3) Ball hitting the pads in line with the stumps OR hitting outside the line of the off-stump with the batsmen offering no shot.
4) Ball would have gone on to hit the stumps.
The above conditions must be met before a batsman can be given out LBW. Now, the final twist in the tale―although it's 'Leg Before Wicket', it's applicable to any part of the batsman's body or cricketing gear (except the gloves). For instance, if a batsman is ducking under a bouncer to avoid being hit, and it hits him on his shoulder (for instance, Sachin Tendulkar's famous dismissal in Australia, although controversial), he can still be given out LBW if the above conditions are satisfied.