Most programming have a set of builtin data types and then ways to combine them to produce more complex data types, for instance,
Primitive Data Types
 Integer
 Float (real)
 Boolean (Logic)
 Character
There may also be some variations on these such as double precision.
The type definitions on this page use a programming language called 'Charity' (http://pll.cpsc.ucalgary.ca/charity1/www/home.html) which takes its ideas from category theory.
Combining Primitive Data Types
We can combine the above builtin types to form compound types in the following ways:
Union
This is where only one of the types applies at a given time.
We might want a type like Shape which could either be a Circle which requires a single float to define it (say radius) or alternatively a Rectangle which requires two floats to define it (say height and width).
type: Shape = Circle float  Rectangle float float
Another possibly might be where we require a function to normally return a numeric value but in case of an error, such as divide by zero, we want it to return a string containing an error message.
A similar concept is enumeration (enum)
Compound (Tuple)
Alternatively the compound type can contain both the data types being combined simultaneously. This is a bit like a fixed length list but where each element in the list could be a different type. 
This page looks at data types as mathematical expressions which we can manipulate in equations.
Recursive Types
We can define compound types recursively. There are two ways to do this:
 Inductive data types, such as list, where the recursion is on the input and the instance on the output of the definition.
 Coinductive data types, such as stream, where the recursion is on the output and the instance on the input of the definition.
inductive  coinductive  

example  type List A = cons:(A,List A)>*  nil:()>*  type Stream A = first:*>(A) & rest:*>(Stream A) 
definition  An inductively defined set is the least solution defined by some relation (inequality). If some other set obeys the relation then it contains the defined set.  A coinductivly defined set is the greatest solution defined by some relation (inequality). If some other set obeys the relation then the defined set contains the coinductivly defined set. 
Algebraic Data Types
Languages such as Haskell and Idris allow types to be defined by listing their possible values, these are the constructors for the type.
These constructors can also be recursive allowing us to define infinite types such as natural numbers and lists.
Types defined in this way allow us to prove things about the type at compile time, especially if the type is defined as a free initial algebra such as natural numbers defined in terms of zero and successor constructors. We can then define plus in a way that we can prove what would otherwise be axioms.
Certain types, such as natural numbers and lists, are fully specified by these constructors. These are known as wtypes [ref 1] . However more complex data types such as groups or graphs would be hard or impossible to fully define in this way although we can still construct something with the required functionality. For example, imagine we want to model a finite group and to do this we want to construct permutations. We could represent a permutation as a list of natural numbers where the value of the nth item in the list represents the index of the entry it maps to. To be a valid permutation the list should contain every index once only but this is not enforced by the constructor.
Perhaps one way to specify this would be to add equations to specify the conditions as in this psudo code:  data perm Nat p : list Nat  p * (inv p) = 1 
This sort of thing can be done in proof assistants but I don't know any general purpose languages with this ability.
Examples of Data Types
List Data TypeThis is inductive: 

Stream Data TypeThis is coinductive: 
Combinatory Logic
For more information about combinatory logic see this page.
Object Oriented
In an object oriented program recursion can be used as follows:
Java  List
In java we encapsulate in an object
class List<E> { E value; List<E> next; }
Scala  Tree
case classes in scala are a halfway between encapsulating in an object and algebraic data types in functional languages.
abstract class Tree case class Sum(l: Tree, r: Tree) extends Tree case class Var(n: String) extends Tree case class Const(v: Int) extends Tree
The case class is similar to  in that it uses pattern matching.
Charity
terminology
1: X > X  identity 
!: Z > 1  terminal map 
<x,y>: W > X * Y  product factorisor 
P0: X * Y > X  first projection 
P1: X * Y > Y  second projection 
C = P0,P1 : X * Y = Y * X  product symmetry 
Δ = <1,1>: X > X * X  product diagonal map 
<vw>: X + Y > W  coproduct factorisor 
b0: X > X + Y  first coprojection 
b1: Y > X + Y  second coprojection 
where:
 x = W > X
 y = W > Y
 v = X > W
 w = Y > W
functor/mapping/function examples:
nil: 1 > C  nil is a function which takes no inputs and returns a single value 
ss: A > C  ss is a function which takesone input and returns a single value 
cons: A * C > C  cons is a function which takes two inputs and returns a single value 
utilities:
list (built in)  data list A > C = nil : 1 > C  cons: A * C > C. 
the "success or failure" datatype: the datatype of exceptions  data SF A > C = ss: A > C  ff: 1 > C. 
the coproduct (sum) datatype  data coprod(A, B) > C = b0: A > C  b1: B > C. 
the boolean datatype (builtin)  data bool > C = false  true: 1 > C. 
the natural number datatype  data nat > C = zero: 1 > C  succ: C > C. 
Inductive Datatypes
bTree  binary tree  data bTree (A, B) > C = leaf: A > C  node: B * (C * C) > C. 
Coinductive Datatypes
triple  data T > triple(A, B, C) = p0_3: T > A  p1_3: T > B  p2_3: T > C. 
conat  data C > conat = denat: C > SF(C). 
colist  data C > colist(A) = delist: C > SF(A * C). 
cobTree  data C > cobTree(A, B) = debTree: C > A + (B * (C * C)). 
Infinite Datatypes
infnat  data C > infnat = poke: C > C. 
inflist  data I > inflist(A) = head: I > A  tail: I > I. 
infbTree  data C > infbTree(B) = infnode: C > B * (C * C). 
Lazy Datatypes
Lid  data Eid A > C = eid: A > C. % eager identify datatype data C > Lid A = lid: C > A. % lazy identify datatype 
Lprod  data C > Lprod(A, B) = fst: C > A  snd: C > B. 
Lnat  data Lnat > N = Lzero: 1 > N  Lsucc: Lid(N) > N. 
Llist  data Llist(A) > L = Lnil : 1 > L  Lcons: A * Lid(L) > L. 
HigherOrder Datatype
exp  The exponetial datatype: the datatype of total functions data C > exp(A, B) = fn: C > A => B. 
proc  * the datatype of total, deteministic processes data C > proc(A, B) = pr: C > A => B * C. * the datatype of partial processes data C > Pproc(A, B) = Ppr: C > A => SF(B * C). * the datatype of nondeteministic processes data C > NDproc(A, B) = NDpr: C > A => list(B * C). 
stack  data S > stack(A) = push: S > SF(A) => S  pop : S > S * SF(A). 
queue  data Q > queue(A) = enq: Q > SF(A) => Q  deq: Q > Q * SF(A). 