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Paraconsistent logic refers to alternative (non-classical) systems of logic which reject the Principle of explosion, which states that once a contradiction has been asserted, any proposition can be inferred from it. Some (but not all) paraconsistent logics are also dialetheic, meaning they hold to a view known as dialetheism, the view that some contradictions are simultaneously true and false.
Does this mental voodoo perhaps destroy the entire point of logic? Not necessarily; some philosophers and scientists assert that true contradictions (dialetheia) exist and therefore should be represented by formal logic. Others, however, hold that these true contradictions are purely semantic or linguisitic in nature & believe that dialetheism can make sense of certain paradoxes. The most notable defender of dialetheism in the contemporary philosophical scene is the British-Australian philosopher Graham Priest. Priest implements the concept of dialetheism in his own system of relevance logic, a particular type of paraconsistent logic.
Paraconsistent mathematics refers to attempts to develop mathematics on top of a foundation of paraconsistent logic and paraconsistent set theory.
Historically speaking, paraconsistency & dialetheism have been common themes in Indian logic, especially Jainist and Buddhist logic. Whereas classical Western logic would see a statement as either true or false, but not both nor neither, Indian logics have traditionally been accepting of statements being both true and false simultaneously, or neither true nor false. Paraconsistency applies in particular to the both case — although it often allows the neither case as well. Paracomplete logic allows the "neither" case by denying the law of the excluded middle.