First of all a Lie algebra is an algebra. That is it is a vector space with a product. The product is a special kind often represented by a bracket [x,y], which takes two vectors and spits out another vector. And it has the key property that [x,y] = -[y,x] for all x and y. So the Lie algebras are all anticommutative. Another property the product has is the Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.
Lie algebras arise from Lie groups. Every Lie group is a manifold and it has a tangent space. And the local tangent space (the fiber) over the group identity is uniquely defined as a Lie algebra. The tangent space is spanned by a set of basis vectors X^a and the product is given by [X^a,X^b] = f^{ab}_cX^c, where the f 's are numbers determined by the group. By studying the Lie algebra you can find out things about the group. Often in physical situations where they have a group (such as a gauge group), they find it easier to work with the Lie algebra.
Lots of standard results about which common Lie groups have which Lie algebras. The group only has one Lie algebra, but the same Lie algebra can come from more than one group.
A lot of any course in Lie algebras consists of ways to classify them. This gets into Dynkin diagrams which are almost magical in some ways -
John Baez just loves them - but which I am NOT going to get into here!
