Overview
Where your database is hosted will depend on what your database is being used for and where your users or application that connects to the database reside.
Location
For a website, the database could reside on the same server if the server provides web and database hosting or alternatively locate the database on a separate server with more disk and memory capacity depending on the website or applications requirements. For large database queries a higher bandwidth on the server may be required and needs to be considered. Another consideration may be one of security, i.e. The database may need to reside on a more secure server than the website or application.
Vendor
Choice of database management system vendor will usually depend on the application's requirements. For a website, MySQL, PostgreSQL or Microsoft SQL Server could suffice. For a large business application, Oracle or Sybase may be the solution. The choice however may be restricted what the 'application' is designed for. Some applications are supplied with a database management built into the install.
Support
If you are running a 24x7 application or website selling goods, you may need to have continuous round the clock support in case of a failure with the database server. Consideration needs to be taken on the impact to your business in the case of database unavailability.
Data
How sensitive is your data? Is it of a financial nature? Does it contain passwords, encrypted data, credit card numbers? This may determine where your host your database, type of server, firewalls employed etc.
Host Providers
There are many good database host providers and one to suit every budget. Some providers are based abroad in the USA and appear to be cheaper. Use caution as their support may be inadequate for your core hours and may consist of expensive international telephone calls. Established providers would be recommended or a local provider who can offer the personal touch and actually attend your company in person should the need arise.