In theoretical physics, cutoff usually represents a particular energy scale or length scale. The phrase denotes the maximal or minimal value of energy, momentum, or length (or something along these lines) such that the objects with even larger (or smaller) values of these physical quantities are ignored.
An infrared cutoff (long-distance cutoff) is the minimal value of energy - or, equivalently, the maximal wavelength (usually a very large distance) - that we take into account.
On the contrary, an ultraviolet cutoff is the maximal allowed energy or the shortest allowed distance (usually a very short length scale). If some quantities are computed as integrals over energy or another physical quantity, these cutoffs determine the limits of integration. The exact physics is reproduced when the appropriate cutoffs are sent to zero or infinity. However, these integrals are often divergent - see IR divergence and UV divergence - and a cutoff is needed.
The dependence of physical quantities on the chosen cutoffs (especially the ultraviolet cutoffs) is the main focus of the theory of renormalization group.