Your question is rather vague but one result that might be of interest is the singular value decomposition theorem which states that every compact operator can be written in the form $$T(x)=\sum \lambda_n (x|x_n)y_n$$ where $(\lambda_n)$ is a sequence of scalars which converges to $0$ and the other two sequences are orthonormal bases for the appropriate Hilbert spaces (which I am assuming to be separable for simplicity). This is a simple consequence of the spectral theorem for compact, self-adjoint operators. Both of these results can be found in virtually any text on Hilbert space or elementary introduction to functional analysis. Googling „Singular value decomposition for compact operators“ will turn up a number of self-contained approaches.

Adding more information based on comments. The connection with the Riesz theorem is, of course, that the latter is the case where the image space is one-dimensional. As pointed out above, one obtains many special classes of operator by assuming that the sequence of scalars is from a suitable sequence space, most notably an $\ell^p$-space ($p=1$ and $p=2$ give the nuclear resp. Hilbert Schmidt classes). Again, as mentioned above, these can be interpreted using tensor products. The class of all continuous linear operators cannot be obtained as a tensor product of Banach spaces (again noted in the comments) but it is a tensor product in the sense of locally convex spaces (Grothendieck) when we regard the Hilbert spaces as complete lcs‘s using the topology of uniform convergence on compact subsets.