Suppose a given alpha diversity metric has value X for a given set of abundances. The effective number of species is then the number of species in a community with equal abundances that would give X for this metric.
This definition can be applied to any metric, regardless of whether it is expressed as a number of species or a more obscure value such as entropy.
Using an effective number of species seems elegant and compelling because it has a natural interpretation and makes all metrics comparable to each other, while metrics using different units such as entropy have no obvious connection to a number of species. However, in practice, abundance distributions are usually highly skeweed with a few very abundant species and a long tail of low-abundance species. With many metrics, the effective number of species is very different from the observed richness, and this can cause the numerical value of the effective number of species to be counter-intuitive.
For example, I found 3,268 OTUs in a soil community had a Shannon entropy of 3.68 bits. With 39 even abundances, the Shannon entropy is 3.66 and with 40 even abundances, the entropy is 3.39, so we can see that the effective number of species should be more than 39 and less than 40. Using the math given in Chao et al. (2010), I found that the effective number of species for a Shannon entropy of 3.66 is 39.7. This indicates that while the community had many OTUs, relatively few high-abundance clusters accounted for most of the reads.
See also Jost's web page discussing the concept of effective number of species.
Chao, Chui and Jost (2010)