All Lie Groups are Lie Groupoids, but not all Lie Groupoids are Lie Groups (in fact a Lie Group can be defined as a Lie Groupoid which has exactly one object).
Although Lie Groups and Lie Algebras can be defined without referring to each other, their relationship (a Lie Group has a corresponding Lie Algebra which is the tangent space at its identity element) is also useful in Maths and Physics. I'm trying to look into the equivalent case for Lie Groupoids with more than one object, which obviously have more than one identity element and so more than one tangent space at their identity elements.
I THINK (but I'm not entirely sure) that a Lie Algebroid is basically a bundle made by 'gluing together' all of these tangent spaces. But I'm not entirely sure that that's right. The more pressing question though, is whether, as this implies, every Lie Algebra satisfies the definition of a Lie Algebroid, just as every Lie Group satisfies the definition of a Lie Groupoid (which has exactly one object). The definitions I've found in Meinrenken et al mean that the definitions of Lie Algebra and Lie Algebroid that I've found are quite different from each other, making it hard for me to figure out if indeed ever Lie Algebra is a Lie Algebroid or not.
Lie algebras are Lie algebroids over a single point. You are right.