What is meant when people say a molecule has high multi-reference character?
When doing a CI calculation based on Hartree-Fock orbitals, then usually the HF configuration has a very high weight in the CI vector (>90%). The next largest weight is then very small (for example only 1%), and there is a very large number of those very small configurations.
Such cases are called weakly or dynamically correlated. Because of the large weight, it is reasonable to optimize the orbitals for the leading configuration only and then run some truncated CI calculations (typically CCSD) on top.

If there are multiple configurations with similar (or same) weight, then it is no longer reasonable to optimize the orbitals for just a single configuration, because this will introduce a bias towards this configuration. The number of these configurations is rather small.
Additionally, those configurations are not necessarily related by Single or Double excitations, so CISD may not include all. However, they are commonly generated by an active space of just a few orbitals.
This is called strong or static correlation, and is typically found in systems with partially occupied close-degenerate orbitals, e.g. d orbitals in transition metals.

In FCI this orbital bias would not matter, because the basis is (numerically) complete. But in truncations like CISD or CCSD this leads to a qualitative error. Some important configurations are poorly represented (due to the bias), other are completely missing (due to the truncation).

A CASSCF calculation resolves the bias by considering all important configurations of similar weight (by chosing an appropriate active space) and balancing the orbitals to all of the considered configurations. Since only some configurations are required for static correlation, it is often feasible to optimize both (CI and orbital coefficients) simultaneously. 

The term multi-reference then refers to a calculation where all (or most) of the CASSCF configurations are used to generate excitations from.