Usage of the term "Free" 
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
 A: (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".
A: The idea of 'free' is that you have some set $S$ (finite or infinite), and you want to create some object (group, module, algebra, etc.) which is both generated by the elements in S, and there are also no restrictions/relations between the elements in S.
So take $S = \{x,y\}$ for example. The free group generated by S would contain $x$ and $y$, but it would also have to contain $xy$, $x^{-1}$, $y^{-1}$, $xyyxy^{-1}x^{-1}y$, etc. And these are all distinct elements. The free abelian group would also contain $x$ and $y$, but since we have the 'abelian' restriction the elements $xy$ and $yx$ are the same. (Abelian group operations are typically written as $+$, so instead we would say $x + y$ and $y + x$ are the same).
One consequence of this is that to describe any homomorphism from the free object generated by $S$ to some other object, it's enough to say where the elements of $S$ go AND there are no restrictions on where the elements of $S$ can go. So for instance the complex numbers are not a free real algebra: $\mathbb{C}$ is generated (as an $\mathbb{R}-$algebra) by $1$ and $i$, but any homomorphism $\mathbb{C} \rightarrow A$ must send $i$ to some element that squares to $-1$. And it's important to know that the notion of 'free' depends on the category being considered. The polynomial ring $\mathbb{R}[x,y]$ is a free abelian $\mathbb{R}-$algebra, but not a free $\mathbb{R}-$algebra.
This last property (that any homomorphism is uniquely described by an arbitrary choice of where the elements of $S$ get sent) is exactly what Thomas Andrews means when he says that 'free' is an adjoint of a forgetful functor.
