What is the difference between a generative and a discriminative algorithm?

Generally, there is a practice in machine learning community not to learn something that you don’t want to. For example, consider a classification problem where one's goal is to assign y labels to a given x input. If we use generative model

p(x,y)=p(y|x).p(x)

we have to model p(x) which is irrelevant for the task in hand. Practical limitations like data sparseness will force us to model p(x) with some weak independence assumptions. Therefore, we intuitively use discriminative models for classification.