DB Connection Handling - I Get "too Many Connection" - What's The Solution?

I have a php-webpage that needs the database in the beginning and in
the end of the script. In the middle there is a lot of processing that
takes several seconds - during that time I don't use the database
connection.

What is the best approach? Should I close the connection after I'm
done with it OR should I reuse the connection thru my whole script?

Where can I read more about this? Any pointers?

Today I'm using the same connection thru my script. And sometimes I
get an error from the database "too many connections". What does that
mean? My webserver provider says that I have at least 100 connections
to play with. How come I get this error?