A SID is the unique identifier for a database, or if it's a RAC system for an instance of a database.
A Service is an identifier for a service offered by the database, and the database should be configured for services such as "BILLING_APP" or "CUST_WEBSITE". http://docs.oracle.com/cd/B28359_01/server.111/b28320/initparams217.htm
Don't confuse a database and an instance, by the way. A database is the set of data and control (etc) files, the instance is the memory areas and processes that access it. RAC has multiple instances per database, each of which can register with the listener to offer all or a subset of the total services offered by the database.
A schema is effectively the username that owns the objects, and there can be multiple schemas in the database.
So an application should be referencing a SERVICE to connect to, through a listener on a certain host and port. The listener resolves this to a database (possibly one of many that register to offer that service). The connection is made to the database as a user who might or might not be the owner of the schema. Typically the connection is not made as the schema owner for security reasons -- in fact the best level of security is to connect as User_A, calling code in User_B's schema, whicxh references tables in other users' schemas.