Hopefully this is a fairly simple question to answer but I've just never gotten around to asking it before. When I start up Windows 98 on some computers it goes straight in and I can start using it. When I start it on other computers however I have to log in. How do I turn off this need to log in? The problem I'm having is that sometimes I plug a network adapter into a computer and have to log on in order to get it onto my network, then I can't turn off the logon.

Search for a file named <username>.PWL on the HD and delete it (or simply search for *.PWL to get a list - there may be more than one user).
The next time when prompted for the password (next boot), leave the password fields blank and press ENTER.
Subsequent boots will not ask for the password anymore.

Aaahhh of course that makes sense. Thanks. Do you know if that will affect my ability to get onto a network?

No, but you will probably have to re-enter the network passwords the first time you connect - if you used to "store" them.
However, if you're not confident, or want to play VERY safe, don't delete the file but rename it to something like <username>.bak or <username>.old so that you can fall back afterwards.
If I remember well, the path is:
C:\WINDOWS\USERNAME.PWL