In many places I have seen linear transformations defined as $[T]^{\alpha}_{\beta}$ where $\alpha$ and $\beta$ are a basis for V and W, respectively.
Another thing I have seen is $[T]^{\alpha}_{\alpha}$ where $\alpha$ is a basis for V.
I have not been able to understand this notation and how to calculate something like this. Could someone please explain what this means.
I'll try to give a somewhat informal account of the situation without going into details.
A linear transformation is a homomorphism of vector spaces, that is a function $T:V\to W$, where $V,W$ are vector spaces, which is linear, i.e. which satisfies
$$T(v+w)=Tv+Tw\text{ and }T(\alpha v)=\alpha Tv$$
for all $v,w\in V$ and all $\alpha$ where $\alpha$ is a scalar of the underlying field.
Now, there is a fundamental theorem in linear algebra which says that for finite dimensional vector spaces, you can always meaningfully represent such a linear transformation as a matrix with coefficients in the field of the underlying vector space.
To fix this matrix representation, one first has to fix corresponding bases for the two spaces. Now, if $\alpha$ is a basis for $V$ and $\beta$ is a basis for $W$, then $[T]_\beta^\alpha$ denotes the matrix representation of $T$ with respect to these bases $\alpha,\beta$ for $V,W$.
The case you mentioned where you've used $[T]_\alpha^\alpha$ is then actually a special case of the above, which applies to the case where $W=V$, so where $T$ is actually an endomorphism, i.e. a linear map between one and the same vector space.
The use of all this is that, by transforming a linear map into its matrix representation, various intrinsic properties of the linear map can maybe be grasp more directly through the simplicity of the matrix as a mathematical object.
Also, for different choices of bases for the underlying spaces, we usually get different matrix representations for a particular linear map and it is of fundamental interest in linear algebra to find bases such that this matrix representation gets a simple as possible.