There are several things to consider when trying to decide whether to place the business logic in the database or in the application code.
Will the same database be accessed
from different websites / web
applications? Will the sites /
applications be written in the same
language or in a different language?
If the database will be used from a single site, and the site is written in a single language then this becomes a non-issue. Otherwise, you'll need to consider the added complexity of stored procedures, triggers, etc vs trying to maintain database access logic etc in multiple code bases.
What are relational databases in
general good for and what is MySQL
good for specifically? What is PHP
best at?
This consideration is fairly straight-forward. Relational databases across the board and specifically in any variant of SQL are going to do a great job at inserting, updating, and deleting data. Generally they also handle ATOMIC transactions well. However, most variants of SQL (including MySQL) are not good at complex calculations, on-the-fly date handling, file system access etc.
PHP on the other hand is very fast at handling calculations, dates, file system accesses. By taking a little time you can even design your PHP code to work in such a way that records are only retrieved once and then stored when necessary.
What are you most familiar /
comfortable with using?
Obviously it tends to make more sense to use the tool with which you are most familiar.
As a last point consider that just because a drill can be used to cut sheet rock or because a hammer can be used to drive a screw doesn't mean that they should be used for these things. Sometimes I think that programmers do more potential damage by trying to make more powerful tools that do everything rather than making simpler tools that do one thing really, really well.