Classical mechanics is that part of classical physics dealing with the deterministic physics of point particle?s and rigid bodies; often the systems with the infinitely many degrees of freedom are also included (like infinite arrays of particles and their continuous limits like classical mechanics of strings, membranes, elastic media and of classical fields). For the continuous systems, the equations of motion can often be explained by the partial differential equations, describing classical physical fields of quantities (typically smooth possibly vector valued functions on manifolds), including background fields like metric; the latter (sub)area is the classical field theory, but it is often studied separately from the classical mechanics of the finite systems of particles; especially if non-classical features or interpretations are involved (e.g. supersymmetry, or unusual case of non-variational equations of motion etc.). In Hamiltonian reduction, due conservation laws, many systems with infinitely many degrees of freedom, reduce to the finite ones.
Edit: I changed the above text, incorporating a part of the discussion (Zoran).
Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.
One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.
Toby: I take your point that ‘dynamics’ was not the right word. But do you draw any distinction between ‘classical mechanics’ and ‘classical physics’? Conversely, what word would you use to restrict attention to particles instead of fields, if not ‘mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)
Zoran: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.
Toby: So ‘mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that.
Nondissipative systems with finitely many degrees of freedom may be described geometrically using symplectic manifolds, or more generally Poisson manifolds; the later may also sometimes appear as reductions of the systems with infinitely many degrees of freedom.
Classical mechanics of a system of point particles and rigid bodies is usually divided into statics, kinematics and dynamics. Statics studies the balance of forces in a system which does not move, or in a stationary flow. Kinematics studies the relation between position, velocity and acceleration of bodies in a mechanical system, without reference to the causes of motion. Dynamics studies motion with reference to the causes of motion and interaction between bodies and its manifestation via (quantified) forces, energy and mass assigned to bodies in motion and interaction.
For a theoretical classical mechanics one often starts with a concrete system of bodies with pulleys, strings, spins, external and internal forces, and dissipative sinks and sources (e.g. friction forces), which are then analysed to get the configuration or phase space of the system, the equations of motion and possibly to determine some special observables of interest. Once abstracted that way, the rest of the study is a rather special case of the theory of dynamical system?s, which itself studies general (either deterministic or stochastic) spatially-parametrized systems in a (discrete or continuous) time evolution.