I have heard the statement that the Higgs coupling is flavor conserving. What does it mean? What kind of coupling would be flavor non-conserving?
Higgs couplings of the Standard Model are flavor conserving because the separate "flavor numbers" such as $L_\mu$ – the number of muons minus the number of antimuons plus the number of muon neutrinos minus the number of muon antineutrinos – are conserved in each interaction.
It means that the corresponding interaction (one that is able to produce the Higgs from no Higgs, or destroy it) vertex either creates a muon-antimuon pair, or annihilates it, or destroys a muon and creates another muon, or destroys an antimuon and creates another antimuon.
This holds for the 3 flavor lepton numbers as well as the 3 generational flavor quark numbers. Note that due to the electric charge conservation, the cubic vertices that conserve the 3 lepton and 3 quark flavor numbers automatically conserve the 6+6 of them, too (the Higgs is neutral).
Interestingly, CMS has detected a 2.5 sigma excess of Higgs bosons that seemingly decay to $$ h \to \mu^\pm \tau^\mp $$ This is an example of a process that would violate the flavor numbers. A muon is created with an anti-tau, or vice versa. Note that this process is compatible with the charge (and energy, momentum etc.) conservation laws, the truly important ones. But it changes $L_\mu$ and $L_\tau $ by $\pm 1$ and $\mp 1$, respectively.
About $1.0\%\pm 0.5\%$ of the Higgs decays seem to be of this flavor-violating kind. It's a significant percentage but the error margin is still compactible with the number being zero. New experiments at the LHC will either confirm this flavor-violating decay or show that the excess was just a fluctuation (due to "luck").