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General principles
Units
Catalytic activity is usually denoted by the symbol z and measured in mol/s, a unit which was called katal and defined the SI unit for catalytic activity since 1999. Catalytic activity is not a kind of reaction rate, but a property of the catalyst under certain conditions, in relation to a specific chemical reaction. Catalytic activity of one katal (Symbol 1 kat = 1 mol/s) of a catalyst means one mole of that catalyst (substance, in Mol) will catalyse 1 mole of the reactant to product in one second. A catalyst may and usually will have different catalytic activity for distinct reactions. See katal for an example.
There are further derived SI units related to catalytic activity, see the above reference for details.

Generic potential energy diagram showing the effect of a catalyst in a hypothetical exothermic chemical reaction X + Y to give Z. The presence of the catalyst opens a different reaction pathway (shown in red) with a lower activation energy. The final result and the overall thermodynamics are the same.
Typical mechanism
Catalysts generally react with one or more reactants to form intermediates that subsequently give the final reaction product, in the process regenerating the catalyst. The following is a typical reaction scheme, where C represents the catalyst, X and Y are reactants, and Z is the product of the reaction of X and Y:

Although the catalyst is consumed by reaction 1, it is subsequently produced by reaction 4, so it does not occur in the overall reaction equation:

X + Y → Z
As a catalyst is regenerated in a reaction, often only small amounts are needed to increase the rate of the reaction. In practice, however, catalysts are sometimes consumed in secondary processes.
The catalyst does usually appear in the rate equation. For example, if the rate-determining step in the above reaction scheme is the first step X + C → XC, the catalyzed reaction will be second order with rate equation v = k_cat[X] [C], which is proportional to the catalyst concentration [C]. However [C] remains constant during the reaction so that the catalyzed reaction is pseudo-first order: v = k_obs[X], where k_obs = k_cat[C].
As an example of a detailed mechanism at the microscopic level, in 2008 Danish researchers first revealed the sequence of events when oxygen and hydrogen combine on the surface of titanium dioxide (TiO₂, or titania) to produce water. With a time-lapse series of scanning tunneling microscopy images, they determined the molecules undergo adsorption, dissociation and diffusion before reacting. The intermediate reaction states were: HO₂, H₂O₂, then H₃O₂ and the final reaction product (water molecule dimers), after which the water molecule desorbs from the catalyst surface.