If the physical laws of a system are time reversal invariant that means essentially that if you were watching a video tape of the system then you would not be able to tell whether it was being played forward or backward. We can represent that concept more mathematically by saying that if a particular motion q(t) obeys the physical laws (e.g. is a solution to the equation of motion) then q(-t) will also obey the physical laws. You could think of this geometrically as reflecting the whole problem across the axis t = 0. If a something is invariant under time reversal it is said to have or obey time reversal symmetry, often denoted T. A system that breaks time reversal symmetry is sometimes said to have an "arrow of time".
An Example of Time Reversal Invariance
A simple example of a system with time reversal invariance is two (idealized) billiards balls colliding on a pool table. Imagine you're playing a game of 8 ball and just the 8 ball is left. You shoot the cue ball at the 8 ball and you aim it dead center, they collide, the cue ball comes to a stop and the 8 ball goes sailing off. Good, now freeze that picture in you mind, and then set it going in reverse. Notice now the 8 ball comes in, collides with the cue ball, comes to a stop, and the cue ball goes sailing off. The time reversed picture looks just as reasonable as the one for forward moving time. In fact, if you didn't know anything about the game and were just looking at the balls, you might assume someone had shot the 8 ball at the cue ball. There are many other examples one can think of that are time reversal invariant, like a mass on a spring or a (perfectly elastic) ball bouncing up and down off the ground.
An Example of Time Reversal Symmetry Breaking
Probably the simplest example I can think of for a system that seems to break time reversal invariance is a box sliding along a level floor. As the box slides, friction slows it down and it gradually comes to a stop. Good, now freeze that picture and play it in reverse. The box is just sitting there and then suddenly it starts moving and keeps speeding up. Now that clearly "just ain't natural". Put more mathematically, that's not a solution to the equation of motion. Similarly, in the example of the bouncing ball, if it's not perfectly elastic then it will bounce to a lower and lower height each time. Again, playing that backward would not look right.
The Role of Time Reversal Invariance in Physics
In the days of classical physics, before Planck, Einstein, and the rest, it was generally thought that microscopic physics was time reversal invariant, and that this was one of the fundamental properties of nature. Elastic collisions, Newtonian gravity, and classical electromagnetism are all time reversal invariant, and these were generally the types of mechanisms that they thought governed the world at the fundamental level. The subsequent development of the standard model of particle physics revealed that nature is not time reversal invariant. Indeed, experiments have been done to measure CP violation, which is equivalent to the violation of time reversal symmetry, T, in the standard model. In that theory CPT symmetry is still upheld, though there are currently experiments looking for violations of that, which would indicate physics beyond the standard model.
If you have been reading closely, you will notice that earlier I gave several examples of simple, everyday situations that seem to violate time reversal invariance, no particle physics necessary. As I said it was thought that microscopic physics was time invariant, but when you talk about macroscopic objects you get into statistical physics and thermodynamics. It turns out that if you talk about a large system and you look with a sort of fuzzy lens that can only really tell what the bulk is doing and not each individual piece, then you lose time reversal symmetry. That is to say that even if the individual pieces obey time reversal invariance, when you look at the whole group in this "fuzzy" way and try to work out its statistical properties, you will get rules that break time reversal symmetry.
Going back to the pool analogy, suppose you are going to break at the beginning of the game. You shoot the pool ball into the large, ordered, triangular group that the rest of the balls make up and they scatter everywhere. Now, if you ran the video tape backward you would see them all coming together, but each collision would obey Newton's laws, nothing fishy yet. On the other hand, if you were looking at this with your "fuzzy" glasses and asking the statistical question, "How likely is it that a scattered group of balls will all come together to form a large, ordered, stationary group with only one (or a few) moving?" the answer would be "Not bloody likely!"
It is these statistical sorts of laws that govern macroscopic objects like an actual rubber ball or billiards ball or a box sliding across the floor. Thermodynamics deals in these sorts of macroscopic, statistical situations and includes the second law of thermodynamics, which says that entropy (which often can be roughly thought of as disorder) in a closed system can never decrease. That law already manifestly breaks time reversal symmetry, because entropy can increase in time but never decrease indicating which direction is "forward" in time. For example, if you take a pot full of hot water and a pot full of cold and put them in contact, thermodynamics says that the hot water will cool and the cold water will warm, which increases the entropy of the system, until they reach equilibrium. You could put a thermometer in each to watch this happen. If you played a video of that in reverse, though, you'd notice it was very strange when one of the two pots started getting hotter and the other colder with no outside influence. This is sometimes called the thermodynamic arrow of time.