One Databse, Multiple Servers: How is it done?

I have code written for a server on a web site such that when a user connects
into it the server accesses a SQL Server database and either adds some
records to the database or returns database information to the user who is
connected. Very straightforward stuff, and it's working, and it properly
handles mutiple connections at the same time using threading and related
techniques.

There is the potential, however, for there to be so many users at any given
time that the single server (hardware) cannot handle it, meaning I need to
add more server computers to the system. I can easily put copies of my server
software on each such server computer, and can even put copies of the
database on each server computer, but that obviously and immediately causes an
issue of keeping all the databases "in synch". As soon as a record is added
to or deleted from a copy of the database on one machine, all the others are
now out of date.

I realize that this is a classic issue, but I have no experience in this
area. Can someone kindly give me a brief overview of how this is typically
resolved, or point me to some online resources that might help me with it?

I'd really appreciate it.

Thank you,

Roger Garrett