Grid computing is where more than one computer coordinates to solve a problem together. Often used for problems involving a lot of number crunching, which can be easily parallelisable.
Cloud computing is where an application doesn't access resources it requires directly, rather it accesses them through something like a service. So instead of talking to a specific hard drive for storage, and a specific CPU for computation, etc. it talks to some service that provides these resources. The service then maps any requests for resources to its physical resources, in order to provide for the application. Usually the service has access to a large amount of physical resources, and can dynamically allocate them as they are needed.
In this way, if an application requires only a small amount of some resource, say computation, then the service only allocates a small amount, say on a single physical CPU (that may be shared with some other application using the service). If the application requires a large amount of some resource, then the service allocates that large amount, say a grid of CPUs. The application is relatively oblivious to this, and all the complex handling and coordination is performed by the service, not the application. In this way the application can scale well.
For example a web site written "on the cloud" may share a server with many other web sites while it has a low amount of traffic, but may be moved to its own dedicated server, or grid of servers, if it ever has massive amounts of traffic. This is all handled by the cloud service, so the application shouldn't have to be modified drastically to cope.
A cloud would usually use a grid. A grid is not necessarily a cloud or part of a cloud.
Wikipedia articles: Grid computing, Cloud computing.
