You'll definitely want to use one. Organizing and storing data is what computers do best. Building your web application around a database is a natural way to build flexibility.
We like Microsoft SQL Server because it offers great performance and all the important features required for OLTP and OLAP at a very reasonable price.
MySQL is an open source alternative, now owned by Oracle, that is gaining in popularity on the Linux, Apache stack.
Oracle and DB2 are comparable, arguably superior, to MS SQL in capability, but are much more expensive for small to medium size applications.
File databases like MS Access are easy and inexpensive, but are severely limited for anything but the smallest use. Having a big company providing upgrades and support for this strategic product is definitely a plus.
Critically important features include standard SQL, full text search, remote administration, stored procedures, error handling, solid security, programming interface, robust multiuser transaction processing and analytics.