There are six quarks and six leptons. Of the gauge bosons, there is one photon, one Z, one graviton, a W+ and a W-, eight different gluons, and at least one Higgs (but there may be several). There is a Higgs-like particle in QCD called the pseudoscalar axion, but not everyone believes this will be manifest as a physical particle.
You might also count the anti-particles of the quarks and leptons, if you like, but the gauge bosons are their own anti-particles (save for the W+ and W-, which complement each other, and the eight gluons, ditto).
It is possible that neutrinos are also their own anti-particles. This was the formulation of Ettore Majorana, as opposed to the Dirac formulation which has both neutrinos and anti-neutrinos. If neutrinos were massless, as was thought until recently, then the two formulations would be mathematically equivalent. It turns out that neutrinos have nonzero masses, however, so someday we will know which formulation is correct.
So how many different fundamental particles are there? It depends how you count them. The string theorists will tell you there's just one.