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A family of pdfs or pmfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$

What is the motivation of this definition? It seems that if would be easier to verify properties of pdfs such as sufficiency if they are of this form? So did people come up with the definition of exponential family after Neyman and Pearson came up with the factorization theorem?

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The WP page answers thoroughly your quesyion, listing as domains of applicability of exponential families to statistics their use in classical estimation, Bayesian estimation, hypothesis testing, and generalized linear models, and explaining why these families are well adapted to these tasks. Which part of your question is not covered by this page? – Did Jan 28 '12 at 8:22
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The Koopman-Pitman-Darmois theorem says that it is only for exponential families that there is a sufficient statistic whose number of components does not grow as the size of an i.i.d. sample grows.

(But that's less than the whole story, I think.)

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