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There are four generally accepted types of paradox. The first is called a veridical paradox and describes a situation that is ultimately, logically true, but is either senseless or ridiculous. A falsidical one presents a problem that usually uses some type of incorrect assumption to justify a result that is, in reality, false. An antinomy or semantic self-referential paradox lays out a set of conditions and then asks a question, the resolution of which becomes self-contradictory, resulting in lack of a valid answer. A dialetheia states that both a statement and the opposite of that statement can both simultaneously be true.
Veridical paradoxes are defined by the fact that the logic applied to a situation is ultimately true within the given context. The most famous example of a veridical problem involves a theoretical man who is 20 years old but has had only five birthdays. The resolution to the problem is that his birthday is on a Leap Day and only occurs once every four years. Although the situation is logically true, the statement is fairly nonsensical.
An example of a falsidical paradox is the idea of an arrow being fired at a target. The exercise assumes that, for the arrow to reach the target, it will have to travel half the distance to get there. Once it is half way toward the target, it must now travel half of the remaining distance to reach the target. Each time the arrow traverses half of the remaining distance to reach the target, it must then travel half of the shorter remaining distance, down to infinitely infinitesimal measurements. This would lead to the conclusion that, since the arrow must always travel half the distance, it would never actually reach the target, which is a false conclusion.
An antinomy presents a statement, question or problem that seems to have no answer according to common sense or a pre-defined set of rules. The barber paradox, a variation of Bertrand Russell's paradox, is an example of this. This antinomy assumes there is a town in which "the barber shaves all and only those men in town who do not shave themselves." The question that is posed is who shaves the barber? If he shaves himself, then he is shaving a man who shaves himself and violates the premise.
Finally, there is the dialetheia. There are no real examples of this type, although there are many philosophical arguments for why they should or should not exist. The general concept is that both a condition and the opposite of the condition can both be true at the same time and co-exist together.