You should really read Wikipedia for in-depth understanding.
In short,
Cloud computing means you develop/run your software remotely on remote platform. This can be either using remote virtual infrastructure (amazon EC2), remote platform (google app engine), or remote application (force.com or gmail.com).
Grid computing means using many physical hardwares to do computations (in the broad sense) as if it was a single hardware. This means that you can run your application on several distinct machines at the same time.
not very accurate but enough to get you started.
