It cannot be stressed enough how ubiquitous Lie algebras
(specifically their representations) are in theoretical
physics. One reason they arise is because for every Lie group
(for example, GL(n,C),
the group of all invertible nxn matrices) one obtains a Lie
algebra by looking at the tangent space at the identity element.
In this way one can reduce problems about representations of Lie groups
(which turn up naturally when we consider the symmetry of physical systems)
to problems about representations of
Lie algebras, for which there is powerful algebraic machinery available.
Definition Let k be a field
(for example the real numbers or complex numbers).
A Lie algebra (over k) is a kvector space g equipped
with a binary operation (x,y) > [x,y]
called the bracket satisfying:

[ax+by,z]=a[x,z] + b[y,z] and
[z,ax+by]=a[z,x] + b[z,y], for a,b in k and x,y,z in g
(bilinearity)

[x,x]=0

[x,[y,z]] + [y,[z,x]] + [z,[x,y]]=0 (Jacobi identity)
Note that the bracket has a skewsymmetric property:
[x,y]=[y,x]
as we can see if we apply the first two axioms to
[x+y,x+y].
A Lie algebra is called abelian if [x,y]=0 for all
x,y in g. Obviously the most interesting ones are
nonabelian. A onedimensional Lie algebra is forced to be abelian
but there is a nonabelian two dimensional Lie algebra. It has
a basis {x,y} such that [x,y]=y.
The most important example of a Lie algebra
is gl(n,k) which is the
collection of all nxn matrices over k.
What is the bracket? For matrices A,B define their bracket
by [A,B]=ABBA. It's not difficult to check that
all the axioms are satisfied and this bracket makes gl(n,k)
into a Lie algebra.
As usual in algebra we think about
substructures, homomorphisms and ideals. A subspace h of a Lie
algebra of g is
a Lie subalgebra if it is closed under the bracket, that is
[x,y] is in h whenever x,y are in h.
A Lie subalgebra is a Lie algebra in its own right.
A function f:g>h
between Lie algebras is a homomorphism if it is a linear transformation
and satisfies f([x,y])=[f(x),f(y)], for all
x,y in g. An ideal h of
a Lie algebra g is a subspace such that [x,y] is in
h for each x in h and y in g.
It's not hard to see that the quotient vector space g/h
gets the structure of a Lie algebra in this case.
The most important examples of Lie algebras occur as Lie subalgebras of
gl(n,k) (in fact a theorem of Ado tells us that if the field
has characteristic zero every finitedimensional Lie algebra turns up
this way). Here are some examples:

sl(n,k) consists of all nxn matrices which have
trace zero, that is the sum of the diagonal elements is zero.
Because the trace of AB is the same as the trace of BA
we see that sl(n,k) is a Lie subalgebra of
gl(n,k).

t(n,k) the collection of upper triangular matrices
and n(n,k), the strictly upper triangular matrices,
are also Lie subalgebras of gl(n,k).
All the examples I gave so far were finitedimensional but there are
also some very interesting infinitedimensional ones.
In general given two vector fields x,y their commutator
xyyx is another vector field and this operation defines
a bracket.
For example
consider g the vector fields on C\{0}.
This space is spanned by z^{n}d/dz,
for n in Z. Defining the bracket as above,
[x,y]=xyyx,
one obtains an infinitedimensional
Lie algebra.
In particular we have
[z^{n}d/dz,z^{m}d/dz]=(mn)z^{n+m1}d/dz
This Lie algebra is called the Virasoro Lie algebra.