Cloud computing refers to a client server architecture where typically the servers (called "the cloud") reside remotely and are accessed via the internet, usually via a web browser. Applications like word processors that have traditionally run locally or on a server and accessed via a dumb terminal are instead run on the remote servers and accessed via a web browser. The same goes for services, such as file storage. Often, the servers are run by a third party and host a set of applications for a variety of clients. One example is Google Docs. Microsoft and Amazon have similar offerings, as do many others.
Grid computing refers to a distributed computing architecture where a set of networked computers ("the grid", typically PCs) are utilized en masse for large computational tasks, typically ones that are embarrassingly parallel. For example, a bank might use such a network to price all their holdings each night. From the point of view of the application doing the calculations, it's just submitting a large number of independent jobs to the grid, and receives the results back. The grid infrastructure handles forwarding each job to a computer, balancing loads, etc.