There's some pretty good answers here but I want to elaborate on all topics:
Cloud: shailesh's answer is awesome, nothing to add there! Basically, An application that's served seamlessly over the network can be considered a Cloud application. Cloud isn't a new invention and it's very similar to Grid computing, but it's more of a buzzword with the spike of recent popularity.
Grid: Grid is defined as a large collection as machines connected by a private network and offers a set of services to users, it acts as a sort of supercomputer by sharing processing power across the machines. Source: Tenenbaum, Andrew.
Cluster: A cluster is different from those two. Clusters are two or more computers who share a network connection that acts as a heart-beat. Clusters are configurable in Active-Active or Active-Passive ways. Active-Active being that each computer runs it's own set of services (Say, one runs a SQL instance, the other runs a web server) and they share some resources such as storage. If one of the computers in a cluster goes down the service fails over to the other node and almost seamlessly starts running there. Active-Passive is similar, but only one machine runs these services and only takes over once there's a failure.