Scaling, or more precisely, scaling invariance is the property of a system near critical point to be invariant under a specific transformation of the system parameters.
Universality in theoretical physics means that in the vicinity of a critical point all the system properties (for instance, thermodynamic properties) exhibit universal behavior: all of them demonstrate power-like dependence with a certain exponent. Each system property has its own exponent. These exponents are called critical indices (or critical exponents). The set of critical indices completely describes behavior of the system near the critical point.
All the model and complicated theories fall into universality classes. Belonging to a given universality class depends on a system properties like a dimensionality of the system, symmetries of the system and etc. All the systems in a given universality class have identical critical indices.
Suppose that it is possible to determine a full set of critical indices for the considered model. It means that all the other problems that associated with the universality class induced by the considered model will be solved.
A number of universality classes is big but it is obvious that a number of different systems exhibiting phase transitions is much bigger.
Due to scaling invariance and universality, two microscopically different systems (for instance, a non-magnetic material and a liquid differ drastically on microscopic scale) can behave identically near the critical point (of course, this point is different for each system). This identical behavior means that all the critical indices coincide.