A stored procedure is a set of SQL commands that can be stored and compiled in the server. After this, the client doesn't have to issue the whole query but can refer to the stored procedure. This gives some more speed because the query only has to be parsed once and there is less data that has to be sent between the server and the client. You can also raise the conceptual level by having libraries of functions in the server.
A trigger is a stored procedure that is invoked when something happens. For example one can install a stored procedure that checks every delete to a transaction table and does an automatic delete on the corresponding customer when all his transactions are deleted.
To see when MySQL might get these functions See section List of things we want to add to MySQL in the future (The TODO)..
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