When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.
The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.
In algebraic topology, a free abstract mathematical object is generated by 
elements in a "free manner" ("freely"),
i.e., such that the 
elements satisfy no nontrivial relations among themselves. To make this more formal,
an algebraic gadget 
is freely generated by a subset 
if, for any function 
where 
is any other algebraic gadget,
there exists a unique homomorphism (which has different
meanings depending on what kind of gadgets you're dealing
with) 
such that 
restricted to 
is 
.
If the algebraic gadgets are vector spaces, then 
freely generates 
iff 
is a basis for 
. If the algebraic gadgets are Abelian groups, then 
freely generates 
iff 
is a direct sum of the integers,
with 
consisting of the standard basis.
