Mulitple Database Connections Vs Single Connection?

I have a DLL that acts as an interface between the application and the database. So you create a new instance to the database via:

MyDatabaseClass db = new MyDatabaseClass()

Each instance of the class creates a new connection to the database.

So if I were to have the following, then 4 connections would be made.

MyDatabaseClass db1 = new MyDatabaseClass()
MyDatabaseClass db2 = new MyDatabaseClass()
MyDatabaseClass db3 = new MyDatabaseClass()
MyDatabaseClass db4 = new MyDatabaseClass()


My question is...What if instead of creating new connections upon each instantiation, I re-use the same connection through a "ConnectionManager" class. In this way, the above code will only create 1 connection instead of 4.

What do you guys think about each approach? Obviously the 2nd approach makes sense for non-web applications, but what about websites? Would that just cause that single connection to be overloaded??

Thanks for any insights.