Hi Anjani,

since the Dirichlet is the conjugate of a multinomial distribution. The posterior distribution of a vector of frequencies in a population, as obtained from a multinomial sample, is Dirichlet. So, the vector of frequencies (porportions) of anything that have multiple states in a population is able to be estimated from a multinomial sample, and would be approached by a Dirichlet density. Examples: gene frequencies when there are more than two alleles present in a population (I have worked with these ones), the proportions of different types of ants (soldier, workers, queens) in a colony, the proportion of the different types of crests in males of a bird species, the porportion of different epidemiological states in a population of mammals (infected, cured, uninfected-vulnerable, uninfected-invulnerable),  the proportion of people with different eyes colors in a population, the frequencies of the nucleotides (A,C,G,T) in a genome as estimated from small fragments of DNA, etc.       

This was really helpful specially the examples. Danke Schon :)