If you are using IE5 you are opening yourself to risks. Upgrade to 6 with all the latest service packs.
As for the dial up network box...
Internet Explorer --> Tools --> Internet Options --> Connections
You will find three settings:
Never dial a connection
Dial whenever an network connection is not present
Always dial my default connection
Best option is the first one, but if you do that then you will have to establish the internet connection manually before you start Explorer or other internet programmes.
As previously mentioned you probably have something running in the background that is trying to contact the Internet. These can come from a variety of sources and even the most careful of us can sometimes unwittingly install them. Lots of freeware programmes now include these nasties. Worst offender is Kazaa/Grokster.
You can get ride of this spyware with a program called Adaware
http://www.lavasoftusa.com/support/download/
Alternatively (or as well) get a firewall package for your PC - Zonealarm is a good one and the basic version is free. The best bit of these programmes is that they will tell you which program is trying to access the internet and will allow you to allow or block the traffic either one-time or permanently.
http://www.zonelabs.com (click on Download & Buy) then click on the Zonealarm link for the free version)
Good luck.
Mike