A category is a set of objects and maps between them along with a composition operator. Suppose A,B,C are objects of a category and f : A -> B, g : B -> C maps then their composite gf : A -> C. composition is associative and there is an identity map for each object.
Let C and D be categories, then a functor F : C -> D is a pair of functions, one mapping objects X,Y,.. of C to objects FX, FY, ... of D, another mapping maps f : X -> Y of C to maps Ff : FX -> FY of D. Functors preserve composition (and therefore identities).
A natural transform eta : K -> H between functors K,H : C -> D has components eta_X : KX -> HX for every object X of C. It satisfies the identity Hf eta_X = eta_Y Kf for every map f : X -> Y.
An adjunction F -| G between functors F : C -> D and G : D -> C is a bijection between maps FA -> B and maps A -> GB which is natural in A and B.
An alternative way to view adjunctions is by applying the bijection to the identity function to get what we call unit and counit natual transforms eta : 1 -> GF, epsilon : FG -> 1. "1" here is the identity functor. Ex. for the reader: Prove the equivalence of the two definitions.
The idea of a monad is to remove any mention of the second category D of an adjunction. Let F -| G be an adjunction and define the endofunctor T = GF : C -> C we have the unit eta : 1 -> T and mu = G epsilon F : TT -> T. Check that these satisfy the monad laws mu Teta = 1, mu etaT and mu muT = mu Tmu.