A paradox is a situation/fact/statement which is surprising because it contains two things which seem to contradict each other (not just surprising). Example: I open the door and I expected to see my cousin - and I actually see a lion. Not a paradox, but certainly a surprise. Giving stimulants to children who are hyperactive makes them calmer. That is a paradox. I can resist anything but temptation. A paradoxical statement which is intended to be humorous. The paradox here lies in the fact that resisting something is definitionally avoiding things which are tempting - so saying you can resist anything but temptation is self-contradictory.

It is the opposite of what you expect, and another way of saying that is "self-contradictory". They usually seem like they would make sense, but when you look at it, it contradicts itself. A good example is "This statement is a lie." If the sentence is a lie, then it must be true, which means it must then be a lie...... Another example happens often in the thought of time travel. "If a man goes back in time and kills his father, then the time traveler would no longer exist, which means there's nobody to kill the father, which means the time traveler exists again." Quite a confusing concept, hopefully this helps!