This is not particularly an answer to your question. Beyond Lie algebroids arising from Poisson structures (which Dima Shlyakhtenko has already mentioned in the comments), I think the Wikipedia list is the standard one. You really should think of Lie algebroids as combining the behavior of (1) bundles of Lie algebras (2) infinitesimal actions (3) foliations.
Instead, I'd like to sketch a maybe better way of thinking about Lie and Courant algebroids and their generalizations. I know very little about the Hitchin+Gualtieri+... picture of geometry. (Although they'll be giving a masterclass together in June in Aarhus, Denmark, so maybe you're planning on going?) So my sketch may or may not have much overlap with their generalizations. By I think it is a dramatic improvement on the very 20th-century approach given on Wikipedia. (Already the definitions I will give are of a very familiar flavor to 20th-century phycisits. The 21st-century improvement is to see Q-manifolds as objects not of a 1-category but of an $\infty$-category, and to think of them as some part (the "derived" part?) of the theory of $\infty$-stacks of manifolds.)
Recall that the category of $\mathbb Z$-graded vector spaces, whose objects are formal direct sums $\bigoplus [n]V_n$ where the $V_n$ are vector spaces (and I use the notation that $[n]$ is a chosen one-dimensional vector space "in degree $n$", and I generally drop $\otimes$ from the notation, so that $[n]V = [n]\otimes V$) has a monoidal structure in an obvious way, and has a non-obvious symmetric structure whereby the braiding $[1]\otimes [1] \to [1]\otimes [1]$ is minus the identity. Let $M_0$ be a finite-dimensional manifold and $V$ a finite-dimensional graded vector space. Then you can form a sheaf, called $\mathcal C^\infty(M_0 \times V)$ of graded-commutative algebras over $M_0$, which assigns to $U \subseteq M_0$ the algebra $\mathcal C^\infty(U) \otimes \widehat{\operatorname{Sym}}(V^\ast)$, where $V^\ast$ is the dual graded vector bundle to $V$, and $\operatorname{Sym}$ is taken in the signed sense, and $\widehat{\operatorname{Sym}}$ denotes completion with respect to the polynomial degree. I will define a $\mathbb Z$-graded manifold to be any sheaf of graded-commutative algebras which is locally isomorphic to a sheaf of the form $\mathcal C^\infty(M\times V)$. In particular, the "total space" of any graded vector bundle is of this type.
The category of graded manifolds is pretty similar to the category of manifolds, and pretty much all of the usual calculus carries over. See for example Rajan Mehta's thesis.
Once we are in graded land, you can talk about fairly right and interesting structures. To make some numbers cleaner, I will use the opposite of the usual convention, and use homological gradings. Then a Q-structure ("Q" is short for "qohomological") on a graded manifold $M$ a vector field $Q$ on $M$ with grading $|Q| = -1$, satisfying $[Q,Q] = 0$, where $[,]$ is the Schouten-Nijenhuis bracket (commutator of vector fields). Notice that this is a nontrivial condition because of the gradings and sign conventions. (More precisely, what is a "vector field with grading $-1$"? Let $M$ be our graded manifold and $\mathrm T M$ its tangent bundle. $M$ also has a trivial vector bundle $[n]\times M \to M$ for each $n \in \mathbb Z$. A vector field with grading $n$ is morphism $[n]\times M \to \mathrm T M$ of vector bundles over $M$.)
Similarly, a degree-$n$ P-structure on $M$ ("P" for "Poisson"; abbreviate "$P_n$-structure") is a degree-$n$ section of $P$ of $\mathrm T^{\otimes 2}M$ satisfying $[P,P] = 0$ (Schouten-Nijenhuis bracket) and either symmetry or antisymmetry depending on the parity of $n$; better is to say that it's a section of $[n] ([-n]\mathrm T)^{\wedge 2} M$ whose corresponding section of $\mathrm T^{\otimes 2}$ satisfies $[P,P] = 0$. Note that, as a section of $\mathrm T^{\otimes 2}$, $P$ defines a map of vector bundles $\mathrm T^\ast \to \mathrm T$; I say that $P$ is symplectic if this map has an inverse $\omega : \mathrm T \to \mathrm T^\ast$.
The most important example: let $A \to X$ be a classical vector bundle. Then the following data are equivalent: (1) a Lie algebroid structure on $A \to X$, (2) a $Q$-structure on the total space of $[1]A$, (3) a $P_1$-structure on the total space of $[1]A^\ast$. Since (1)->(2) is covariant, I like to take (2) as the definition of (1); it explains what are the correct morphisms of Lie algebroids. (Actually, the category of classical Lie algebroids is only a full subcategory of the category of Q-manifolds; it is full on those manifolds whose algebras of functions are generated in gradings $0,-1$, so that the "geometric" gradings of the manifold are precisely $0,1$.)
Another important example: the total space of $[n]\mathrm T^* M$ is $P_n$ (symplectic) for any graded manifold $M$. When $n = 0$, this is the symplectic structure you already know; when $n=\pm 1$, the P structure is the Schouten-Nijenhuis bracket.
Anyway, one can string adjectives together, talking about e.g. $QP_n$-manifolds, which would be a manifold with a vector field $Q$, a bivector field $P$, satisfying $|Q| = -1$ and $|P| = n$, and satisfing $[Q,Q] = 0$, $[P,P] = 0$, and $[Q,P] = 0$. Or you could have two (commuting) Q-structures. And so on.
If memory serves, a Courant algebroid is the following (the best reference is Roytenberg's thesis), in the same way that a Lie algebroid is a Q-manifold (so as with Lie algebroids, Courant algebroids are the following plus restrictions on the gradings of the generators). A Courant algebroid is a $QP_2$ manifold whose Poisson structure is symplectic. Important examples include $[2]\mathrm T^\ast M$ for $M$ a Q-manifold, and in particular $[2]\mathrm T^\ast [1]A$ for $A \to X$ a Lie algebroid (extend the Q-structure on $M$ to a Q-structure on $[2]\mathrm T^\ast M$ by taking Lie derivatives in the $Q$-direction). But more generally: if $M$ is a $P_n$ manifold, then the Poisson structure on $M$ can be encoded as a Q-structure on $[n+1]\mathrm T^\ast M$. So if $N$ is $QP_1$, then $[2]\mathrm T^\ast M$ has a Q-structure from $Q|_M$, a Q-structure from the $P_1$-structure on $N$, and they commute, so add them and get a $Q$-structure on $[2]\mathrm T^\ast N$, and it still commutes with the canonical symplectic structure. (You can tell apart the two $Q$ structures if you remember the euler vector field that encodes that $[2]\mathrm T^\ast N$ is the total space of a vector bundle over $N$.) Finally we get the main example: if $A\to X$ is a classical Lie bialgebroid, then $[1]A$ is $QP_1$, so $[2]\mathrm T^\ast[1]A$ is Courant. (Note that there exists an isomorphism $[2]\mathrm T^\ast[1]A \cong [2]\mathrm T^\ast[1]A^\ast$ of $QP_2$ manifolds over $X$, and that $M = [2]\mathrm T^\ast[1]A$ as a graded vector bundles over $X$ is of the form $ M \cong [2]\mathrm T^\ast X \oplus [1] (A\oplus A^\ast)$, and so you can give equivalent data to $M$ by giving $A\oplus A^\ast$ some interesting structure, which is the usual way of defining/building Courant algebroids.)
One final comparison: a $QP_1$ manifold whose P-structure is symplectic, if its algebra of functions is generated in non-positive gradings, is necessarily of the form $[1]\mathrm T^\ast X$ for $X$ a classical manifold, and the Q-structure exactly encodes a Poisson structure on $X$. So the usual definition of Courant algebroid is: what I gave above, and ask that the algebra of functions is supported only in nonpositive gradings --- i.e. the manifold itself has geometric gradings only in nonnegative gradings. Then it's clear that Courant algebroids are a generalization of Poisson manifolds. In the other direction, a $QP_0$ symplectic manifold with the same grading restrictions is necessarily a classical symplectic manifold with $Q=0$. Symplectic, Poisson, Courant, .... But from a graded-geometry point of view, restricting the gradings of the manifold is somewhat unnatural. You should instead see the sequence Symplectic, Poisson, Courant, ..., as being a sequence of full subcategories of some much richer categories.
I've mentioned already Mehta and Roytenberg as good references. Of course, canonical references also include Weinstein and collaborators, Severa, Cattaneo+Felder, anything with the string "AKSZ" in the title (including the original paper by A,K,S, and Z), and probably many others.
