First, I'll start my answer by defining trigger: a trigger is an stored procedure that is run when a row is added, modified or deleted.
Triggers can run BEFORE the action is taken or AFTER the action is taken.
BEFORE triggers are usually used when validation needs to take place before accepting the change. They run before any change is made to the database. Let's say you run a database for a bank. You have a table accounts and a table transactions. If a user makes a withdrawal from his account, you would want to make sure that the user has enough credits in his account for his withdrawal. The BEFORE trigger will allow to do that and prevent the row from being inserted in transactions if the balance in accounts is not enough.
AFTER triggers are usually used when information needs to be updated in a separate table due to a change. They run after changes have been made to the database (not necessarily committed). Let's go back to our back example. After a successful transaction, you would want balance to be updated in the accounts table. An AFTER trigger will allow you to do exactly that.
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