I assume 'F' stands for functor.
Signature of an Algebra
We can think of a binary operation as taking two numbers and returning another number. So
a = b * c
is a mapping
(b,c) > a
Since a,b & c are all of the same type (such as real numbers) we can show the type signature as:
T > T²
where
 T² = T×T
 × = the cartesian product
Note: we have reversed the direction of the functor, we will see later that for an algebra the direction of the operation and the functor are reversed and for a coalgebra they are not.
So in the case of numbers, T is the set of all numbers so T² is the cartesian product of these sets.
So far we have just looked at one operation in an algebra, lets now look a whole algebra, an example might be groups. A group has one binary operation (multiplication), one unary operation (inverse) and one identity element. The binary operation has a domain type of T², the unary operation has a domain type of T and the identity has a domain type of 1. We can therefore give the signature functor F of an algebra as a polynomial:
F : T > T² + T + 1
where
 T² = T×T
 × = the cartesian product
 + = disjoint union
 1 = identity element (final object)
So in the reverse direction we can get functors like these:

1 > T 



T > T 



T² > T 

FAlgebra & FCoalgebra
The functor 'F' that we have defined above is an endofunctor to/from a category C which contains the type T and its powers to allow the following:
FAlgebra  FCoalgebra 

An FAlgebra consists of a pair: (T, α) which is a type and a function α which is defined as: α : F T > T 
An FCoalgebra consists of a pair: (T, β) which is a type and a function β which is defined as: β : T > F T 