BEGIN initiates a transaction block, that is, all statements after BEGIN command will be executed in a single transaction until an explicit COMMIT or ROLLBACK is given. By default (without BEGIN), PostgreSQL executes transactions in "autocommit" mode, that is, each statement is executed in its own transaction and a commit is implicitly performed at the end of the statement (if execution was successful, otherwise a rollback is done).
Statements are executed more quickly in a transaction block, because transaction start/commit requires significant CPU and disk activity. Execution of multiple statements inside a transaction is also useful to ensure consistency when making several related changes: other sessions will be unable to see the intermediate states wherein not all the related updates have been done.
START TRANSACTION has the same functionality as BEGIN.
Issuing BEGIN when already inside a transaction block will provoke a warning message. The state of the transaction is not affected.
BEGIN is a PostgreSQL language extension. There is no explicit BEGIN command in the SQL standard; transaction initiation is always implicit and it terminates either with a COMMIT or ROLLBACK statement.
Other relational database systems may offer an autocommit feature as a convenience.
Incidentally, the BEGIN key word is used for a different purpose in embedded SQL. You are advised to be careful about the transaction semantics when porting database applications.