MySQL replication stops whenever there is an error running a query on the slave. This happens so the problem query can be identified and resolved.
Such errors can be skipped as long as you know why the query failed.
For example, when you run a query in the slave accidentally instead of running it on the master. You can skip just that one query that is hanging the slave using:
mysql> SET GLOBAL SQL_SLAVE_SKIP_COUNTER=1;
mysql> START SLAVE;