In general terms, inbreeding is dangerous because it increases the chances of homozygosity. This means that for a particular gene, identical alleles of the gene are present on both homologous chromosomes. Those individuals result of the inbreeding have more chances of having recessive or deleterious traits.
These are generally 'invisible', they don't express physically, but they are present in the genes so if a person has two of them, they can become a manifest trait. Some examples of negative traits are reduced fertility, increased genetic disorders, lower birth rate, higher infant mortality or loss of immune system function.
You can calculate the risk of recessive traits (an example of inbreeding calculation for dogs, and more info here). It is greater when the parents are close relatives and lower for relationships between more distant relatives:
A measure of inbreeding of an individual A is the probability F(A)
that both alleles in one locus are derived from the same gene in an
ancestor. This probability F(A) is called the
"coefficient of inbreeding".
Both the inbreeding and the coancestry coefficients can be defined for specific individuals or as average population values, but they assume no selection or are limited to neutral alleles, so the numbers are a simple statistical calculation.
If you want to know how inbred a population is, you can use the effective population size. You can note how the average inbreeding coefficient changes from one generation to the next, and then define Ne as the size of the idealized population that has the same change in average inbreeding coefficient as the population under consideration.