Convolution in the frequency domain
the FFT is a fast method of doing the DFT.    yes, multiplication in one domain corresponds to convolution in the reciprocal domain.  and since the DFT is a totally circular operation, any consequential convolution done in one domain by the DFT by multiplication in the other domain is circular convolution.  you don't have a choice about it.  to make a circular convolution look like linear convolution, you must zero-pad sufficiently.

how many zeros to pad?  need to consider how fast convolution (overlap-add or overlap-save) works.  if $N$ is the DFT length, the non-zero lengths of the two sequences being convolved, must not add to more than $N+1$ otherwise there will be overlapping and aliasing of non-zero components.