If you make the opening shot in billiards, the balls are all over the table in the end. In principle you could now make another shot which restores the original, ordered position, but in practice it's impossible. So the opening shot is called 'irreversible'.

Same thing goes for many things, for example letting some hydrogen gas out of a gas bottle: The gas will never go back into the bottle by itself (even if there is a very very small probability that this may happen, in practice it never will). Because the behavior of this kind of systems is described in thermodynamics those systems are called "thermodynamically irreversible".

In the end it simply means that its much more easy to break things than to assemble them again (which also leads to the concept of the always increasing entropy).