The String Coffee Table

In the context of categorification, algeboids show up as one member among the family of $k$-tuply stabilized Lie $n$-algebras. As discussed in

J. Baez & A. Crans
Higher-dimensional Algebra VI: Lie 2-Algebras
math.QA/0307263

these fit into an infinite table which begins like this:

It is easy to give a nice definition of a groupoid: this is just a category in which all morphisms are isomorphism. In other words, this is a set of objects over each of which lives a group $G$ and which are mutually connected by group automorphisms in $\mathrm{Aut}(G)$. If the space of morphisms and objects as well as the source and target maps between them are all smooth, we have a Lie groupoid. One example is the Lie groupoid of smooth paths in some manifold.

A Lie algebroid should be the infinitesimal version of a Lie groupoid just like a Lie algebra is the infinitesimal version of a Lie group. However, it seems that nobody knows a similarly nice way to define an algebroid, really, and so the standard definition is instead the following:

A Lie algebroid is a vector bundle

over a manifold $M$ together with a Lie algebra structure $[.,.]$ on the space $\Gamma(A)$ of smooth sections of $A$, and a bundle map $a : A \to TM$ (called the anchor), extended to a map between sections of these bundles, such that

i) $a([X,Y]) = [a(X),a(Y)]$

ii) $[X,fY] = f[X,Y] + (a(X)f)Y$

for any smooth section $X$ and $Y$ of $A$ and any smooth function $f$ on $M$.

So in other words this is a vector bundle over $M$ with a Lie bracket on its sections which has a representation on the space of vector fields over $M$.

You can find this definition for instance in

Ping Xu
Gerstenhaber algebras and BV-algebras in Poisson Geometry
dg-ga/9703001

As the above table already suggests, there is a web of interesting relations of algebroids to all kinds of special algebras like differential algebras, Gerstenhaber algebras and BV-algebras.

In fact, maybe surprisingly at first sight, the above definition of the Lie algebroid $A$ is equivalent to saying that the space of smooth sections of the exterior power bundle $\Lambda A^*$ is a differential graded algebra. The proof for this essentially amounts to realizing that this is the dual of the statement above in that we go from the differential graded algebra of the exterior cotangent bundle of $M$ to a more general vector bundle.

One of the more important examples for Lie algebroids are those coming from Poisson manifolds $M$ with Poisson tensor $\mathcal{P}$. Here we can choose the bundle $A = T^*M$ as the cotangent bundle of $M$ and the anchor map

Now, there is a famous 2-dimensional $\sigma$ model with target space $M$ a Poisson manifold known as the Poisson $\sigma$-model. Its field content is the embedding map

and a covector-valued worldsheet 1-form $\alpha$, which together can be thought of as defining a vector bundle morphism

where, since the target is Poisson, we can think of both of these vector bundles as Lie algebroids of the above type.

The action functional $S$ is given by

and it turns out that the equations of motion following from this say that $\phi$ is not just a vector bundle morphism but in fact a Lie algebroid morphism respecting the algebroid structure.

Thomas Strobl is an expert on this stuff and has lots of papers on this topic. The above for instance can be found in the introductory sections of

M. Bojowald, A. Kotov & Th. Strobl
Lie algebroid morphisms, poisson Sigma Models, and off-shell closed gauge symmetries
math.DG/0406445

This is related to all kinds of things like Chern-Simons theory, topological strings, generalized complex geometry. Maybe I can mention some of that later, but since I am running out of time for now all I will do is refer to the general literature.