An terminal object has a unique morphism from every other object in the category.
Terminal objects give a category theory version of the concept of 'element' in set theory. 1 > A allows us to pick out an arbitrary element of the set A.
Terminal Object in SetAlthough category theory is about the external properties of objects, to get an intuitive understanding, it can be helpful to peek inside them. In setlike categories the terminal object is like the single element set. In set, maps exist from any nelement to an melement set (provided m≠0). So every set has maps to every nonempty set, however only maps to the 1element set are unique. 
Examples in Various Categories
Terminal Object  

Notation  1  
generalisation  a kind of limit  
universal cone over diagram  empty diagram 

examples: set:  {1}or {a} ... set with one element (singleton) 

group (null object)  trivial group (just identity element)  
topological space  single point  
poset  greatest element (if exists)  
monoid  trivial monoid (consisting of only the identity element)  
semigroup  singleton semigroup  
Rng  trivial ring consisting only of a single element 0=1  
fields  does not have terminal object  
Vec  zero object  
Top  onepoint space  
Grf  graph with a single vertex and a single loop  
ΩAlg algebra with signature Ω 

Cat  category 1 (with a single object and morphism) 