I know that a regular matroid is one that can be represented by a totally unimodular matrix.
I also know that a rank r totally unimodular matrix is a matrix over $\mathbb R$ for which every submatrix has determinant in $\{ 0, 1, -1 \}.$
But I still do not understand what does it mean to find the number of bases of regular matroid. Are we counting the number of totally unimodular matrices that can represent this regular matroid? If so, how can I interpret this sentence in terms of the definition of a totally unimodular matrix? If not, then can someone explain to me the correct meaning please?
Bases in matroids are actually defined to generalize bases in linear algebra, i.e., bases of vector spaces. Recall in that context, a basis is a set of linearly independent elements that span the space. A basis $B$ of a matroid is a set of independent elements that span the matroid, i.e., $r(B) = r(M)$.
When we say a matroid $M$ is represented by a matrix $A$, we mean that this correspondence is exact, i.e., the elements $M$ are in one-to-one correspondence with the columns of $A$, and that a subset of $E(M)$ is independent if and only if the correspondent subset of columns is linearly independent. The rank also $M$ also corresponds to the rank of $A$. Thus, to find the number of bases of a matroid is equivalent to finding the number of bases of $A$.
Note that nothing about is specifically about totally unimodular matrices/regular matroids, this holds for representable matroids, of which regular matroids are a special subclass. Representable matroids are matroids for which there exists some field over which we can find this matrix correspondence. It turns out that regular matroids are representable over every field, which is amazing. This is sometimes taken to be a definition, in fact.