This page describes some examples of adjunctions. Adjunctions were introduced on the page here.
Example  Monoid
On this page is an example of an adjunction between set and a free monoid (List monad).
Example  Preordered Set
See this page for preordered set example.
Sum and Product
+Δ×
The top triangle is the counit: The lower triangle is the unit: 

The top triangle is the counit: The lower triangle is the unit: 
This is explained by Bartosz Milewski on his blog here.
We can also relate this to logic:
existential,sum Σ  weakening 

weakening  universal, product Π 
Currying
This relates product to functions.
Some Other Very General Adjunctions
equality  contraction  
truth  comprehension or subset types  
equality  comprehension  
quotients  equality  
Adjoint Pairs
The basis on a vector space
The free group on a set
G G / [G,G] commutator subgroup
universal enveloping algebra of a Lie algebra
completion of a metric space