TITLE: What is a coalgebra intuitively?
QUESTION [47 upvotes]: How to think about coalgebras? Are there geometric interpretations of coalgebras? 
If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic examples of coalgebras should one keep in mind?
Anything that helps to think about coalgebras without headache is welcome ;)

REPLY [6 votes]: a pretty general kind of coalgebra is the path coalgebra of a quiver (directed graph).