Two-dimensional stellar kinematics of 48 representative elliptical (E) and lenticular (S0) galaxies obtained with the SAURON integral-field spectrograph 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 , which involves luminosity-weighted averages over the full two-dimensional 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, depending on whether they have λR values within their effective radius Re below or above 0.1, respectively. Slow and fast rotators are shown to be physically distinct classes of galaxies, a result which cannot simply be the consequence of a biased viewing angle. Fast rotators tend to be relatively low-luminosity galaxies with MB >~ -20.5. Slow rotators tend to be brighter and more massive galaxies, but are still spread over a wide range of absolute magnitude. Three slow rotators of our sample, among the most massive ones, are consistent with zero rotation. Remarkably, all other slow rotators (besides the atypical case of NGC 4550) contain a large kpc-scale kinematically decoupled core (KDC). All fast rotators (except one galaxy with well-known irregular shells) show well-aligned photometric and kinemetric axes, and small velocity twists, in contrast with most slow rotators which exhibit significant misalignments and velocity twists. These results are supported by a supplement of 18 additional early-type galaxies observed with SAURON. In a companion paper (Paper X), we also show that fast and slow rotators are distinct classes in terms of their orbital distribution. We suggest that gas is a key ingredient in the formation and evolution of fast rotators, 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. Detailed numerical simulations in a cosmological context are required to understand how to form large-scale KDCs within slow rotators, and more generally to explain the distribution of λR values within early-type galaxies and the distinction between fast and slow rotators.