The difference between WFE, Application Server and Database Server in a SharePoint Farm

The Web Front end Server is the server that is hosting IIS and serves the websites.
An Application Server  has services on it, for example, like Excel services, Project server, etc..
A database server simply hosts the SharePoint core db's and the other Content databases.. It just has SQL Server installation on it.

So you have to install Sharepoint to WFE's and App. Servers.. You dont need to install Sharepoint to the database servers on your farms.