Zero-inflated distributions are used to model count data that have many zero counts. For example, the zero-inflated Poisson distribution might be used to model count data for which the proportion of zero counts is greater than expected on the basis of the mean of the non-zero counts.
In recent years, there has been considerable interest in regression models based on zero-inflated distributions. Much of this interest stems from the seminal paper of Lambert [Ref 1], though this type of model appears to have originated in the econometrics literature.
Clarice Demetrio, John Hinde and I wrote a review paper on models for count data with many zeros for the International Biometric Conference, Capetown, December 1998 [Ref 2]. Although this review was reasonably complete at the time, there is a substantial subsequent literature on this topic, including one contribution of our own [Ref 3].
 Lambert, D. (1992) Zero-inflated Poisson regression, with an
application to defects in manufacturing. Technometrics, 34,
 Ridout, M.S., Demetrio, C.G.B. and Hinde, J.P. (1998) Models for counts data with many zeros. Proceedings of the XIXth International Biometric Conference, Cape Town, Invited Papers, pp. 179-192. [pdf]
 Ridout, M.S., Hinde, J.P. and Demetrio, C.G.B. (2001) A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives. Biometrics, 57, 219-223. [journal link]