Well, familiarity depends on your level of involvement. So look for the definition of <rank> in your notes/textbook. You can't find it (though i doubt it), search the internet: planetmath.org, wikipedia.org, wolfram sites offer a wealth of information on linear algebra.
m is the dimension of the vector space V. It's maximum number of linear independent vectors in V.
Of course not, you have to know that from the beginning of the class on linear algebra. The range is a subset of V obtained by taking all possible images through T of all the elements of U (assumed to be equal to the domain of T). The range is also known as codomain of a linear map/operator.
If U is a linear space and T is a linear operator, one can easily show that range(T) is also a linear space, namely a vector subspace of V.