What kind of operations are involved in NTRU? I've read that lattice based algorithms involve matrix-vector products. Is this the case of the NTRU algorithm?

When I've read the details of the NTRU algorithm, I've seen products of polynoms. Where are the matrix-vector products?
Almost all lattice-based schemes that have somewhat practical performance use lattices from a subset of all lattices. Those lattices have additional structure and can be described using polynomials. The big advantage of doing this is that on the one hand the size of a description of the lattice is much smaller than for arbitrary lattices and on the other hand that one can use the fast Fourier transform (FFT) to do multiplication. If you want to learn more about this google for Ideal lattices or "the Ring setting" / Ring-LWE. 

A different point one should note is that these advances in terms of performance are caused by additional structure that these lattices provide. Currently, we do not know any attacks that can exploit this additional structure to actually break the schemes (at least for the classes of lattices proposed). However, this is a topic of ongoing research and it is already known that for some spacial cases (if one selects these subsets in a bad way) schemes get insecure (see e.g. https://eprint.iacr.org/2015/176.pdf).