TITLE: Usage of the term "Free"
QUESTION [5 upvotes]: What is the difference between free groups and free modules/vector/algebras spaces, or in other words what does free mean in algebra?

I have seen two uses of the term free:

something of infinite order that is generate by a linear independent set called a basis. I have seen this applied to (Groups/abelian groups/vector spaces/modules/algebras)
an algebraic structure in which there is no defined relation between elements. I have seen this apllied to vector spaces/modules-used in tensor product definition.

Thanks

REPLY [7 votes]: (This is essentially a down-to-earth translation of Thomas Andrews comment)
If a foo is a set with certain additional structure, then a free foo  over a certain set $S$ of generators ( or the foo freely generated by the set $S$) is a foo $F$ with the following properties:

$S$ is a subset of the underlying set of $F$.
Whenever $G$ is a foo and $f\colon S\to G$ is a map from $S$ to the underlying set of $G$, there exists one and only one foo-homomorphism $\phi\colon F\to G$ such that the restriction of $\phi$ to $S$ (viewed as a set-map and not a homomorphism) equals $f$.

To go into more detail what "underlying set" and "homomoprhism viewed as set-theoretical map" mean, one should learn about category theory, where thse concepts are introduced via "forgetful fucntors".