Informally, a free functor is a left adjoint to a forgetful functor – part of a free-forgetful adjunction. (This is informal because the concept of forgetful functor is informal; any functor might be viewed as forgetful, so any left adjoint might be viewed as free, while in practice only some are.)
Formally, with respect to a monad or algebraic theory or operad $T$, and $T Alg(C)$ the corresponding category of algebras over a monad or algebras over an algebraic theory or algebras over an operad, respectively, in some category $C$, the free $T$-algebra functor is the left adjoint to the forgetful functor $T Alg(C) \to C$.
Such a functor may be thought of as sending any object of $C$ to the $T$-algebra freely generated by it.
In general, if $U: C \to D$ is thought of as a forgetful functor and $F: D \to C$ is its left adjoint, then $F(x)$ is the free C-object on an object $x$ of $D$.
More generally, even if the entire left adjoint $F$ doesn’t exist, a free object on $x$ can be defined using a universal property, as “what the value of $F(x)$ would be if $F$ existed.” Conversely, if a free object on $x$ exists for all $x\in D$, then the left adjoint $F$ can be assembled from them.
Dually, a cofree functor is a right adjoint to a forgetful functor.
For the classical functors which forget algebraic structure, cofree functors are less common than free functors. As a political joke (which works best for someone who associates political freedom with the left wing), cofree functors have sometimes been called fascist functors. Some discussion of this joke may be found at the nForum.
Classically, examples of free constructions were characterized by a universal property. For example, in the case of the free group on a set $X$ the universal property states that any map $X \to G$ as sets uniquely extends to a group homomorphism $F(X) \to G$. When such a free construction can be realized as a left adjoint functor, this universal property is just a transliteration of the fact that the unit of the free-forgetful adjunction is an initial object in the comma category $(X \downarrow \operatorname{for})$ (see e.g. the proof of Freyd’s general adjoint functor theorem.)
the free monoid functor $Set \to Mon$;
the free module functor $Set \to K Mod$ for a rig $K$;
the free group functor $Set \to Grp$;
the free abelian group functor $Set \to Ab$;
the abelianization functor $Grp \to Ab$;
the free category functor $Quiv \to Cat$;
the free operad functor;
the unitisation functor $Rng \to Ring$.
One formal sort of free functor is the left adjoint $C\to C^T$, where $T$ is a monad on the category $C$ and $C^T$ is its Eilenberg-Moore category (the category of $T$-algebras). This includes all of thee examples above and many others.
A general way to construct free functors is with a transfinite construction of free algebras (in set-theoretic foundations), or with an inductive type or higher inductive type (in type-theoretic foundations).
The cofree coalgebra on a vector space. More generally, if $M$ is an operad in a symmetric monoidal category $V$, $Prop(M)$ its associated PROP, and if $C$ is a monoidal $V$-category, then an $M$-coalgebra in $C$ may be identified with a monoidal $V$-functor $Prop(M)^{op} \to C$. Under suitable completeness assumptions on $C$, the forgetful functor $M$-$Coalg_C \to C$ has a right adjoint, and this forgetful functor is comonadic.
If $M$ is a monoid, the forgetful functor $Set^M \to Set$ on (left) $M$-sets has a right adjoint $X \mapsto \hom(M, X)$, where $M$ acts on functions $f: M \to X$ according to the rule $(m f)(m') = f(m' m)$. This forgetful functor is comonadic. Much more generally, the right adjoint to the underlying functor $Set^C \to Set/C_0$ ($C_0$ the set of objects of a category $C$) is comonadic. More generally still, if $V$ is complete and $f: C \to D$ is a functor between small categories, the functor $V^f: V^D \to V^C$ has a right adjoint (although $V^f$ will not normally be comonadic in this generality).
The forgetful functor $Cat \to Set$, taking a small category to its set of objects, has a right adjoint $K$ for which $K X$ is a category whose objects are elements of $X$ and where there is exactly one morphism $x \to y$ for any $x, y \in X$. The category $K X$, which is a groupoid, is known as the chaotic category on $X$, or the indiscrete category on $X$.
When $U: C \to Set$ is topological concrete category over $Set$, as for example the forgetful functor $U: Top \to Set$, it frequently happens that $U$ possesses a right adjoint, assigning to a set an “indiscrete topology”.
The ring of Witt vectors is the co-free Lambda-ring.
A rich source of examples is coreflective subcategories, which are comonadic over the ambient category. For example, the category of compactly generated spaces is coreflective in the category of all spaces, $Top$.