2D stellar kinematics of 48 representative E and S0 galaxies obtained with the SAURON IFS reveal that early-type galaxies appear in two broad flavours, depending on whether they exhibit clear large-scale rotation or not. We define a new parameter LambdaR which involves luminosity weighted averages over the full 2D kinematic field, as a proxy to quantify the observed projected stellar angular momentum per unit mass. We use it as a basis for a new kinematic classification: early-type galaxies are separated into slow and fast rotators (SRs, FRs), depending on whether they have LambdaR values within their effective radius Re below or above 0.1, respectively. SRs and FRs are shown to be physically distinct classes of galaxies, a result which cannot simply be the consequence of a biased viewing angle. FRs tend to be relatively low luminosity galaxies. SRs tend to be brighter and more massive galaxies, but are still spread over a wide range of absolute magnitude. 3 slow rotators of our sample, among the most massive ones, are consistent with zero rotation. Remarkably, all other SRs contain a large kpc-scale KDC. All FRs show well aligned photometric and kinemetric axes, and small velocity twists, in contrast with most SRs which exhibit significant misalignments and velocity twists. In a companion paper (Paper X), we also show that FRs and SRs are distinct classes in terms of their orbital distribution. We suggest that gas is a key ingredient in the formation and evolution of FRs, and that the slowest rotators are the extreme evolutionary end point reached deep in gravitational potential wells where dissipationless mergers had a major role in the evolution, and for which most of the baryonic angular momentum was expelled outwards. (abridged)
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