best practice in upgrading db structure

This is going to be an amateur question...

Could somebody explain me, or point me to a resource where I can find
out what is the recommended practice when a live db needs to be replaced
with a new version of it that has a slightly different structure?

My first guess would be to create the empty new version of the db and
insert the data of the old one into it - adding and modifying it when
necessary, but I am not sure.

What do you usually do in a situation like this?

Thanks for the help.
Balázs