The short answer
For the impatient, here's the simple answer:
:\> regsvr32 jscript.dll
The longer answer
That's not the only possible issue that can cause this. For the particular computer I was fixing, it was the jscript.dll file that needed re-registering but it might not be the case for you. Microsoft has an excellent KB article with numerous suggested fixes that covers the relevant bases nicely. I'll just mention them here briefly:
Use Compatibility Mode - If it's a problem with IE8 and doesn't appear in older versions then it might be that the website hasn't caught up to more recent changes in IE.
Verify Time/Date settings - Lots of security related software are sensitive to date/time settings. If you can't login to websites with authentication then it's worth a check.
Clear SSL state - I didn't even remember this option until I read the KB article. Anything that stores state info can experience the occasional corruption no matter how good it is.
Re-register .dll files -It's basically the same solution as the above section but Microsoft lists a different set of .dll files that may be causing problems.
Reset Internet Explorer - IE7 introduced the option to reset the whole browser to "factory settings". It comes in handy for fixing quite a lot of browser problems, particularly those to do with add-ons.
Use System Restore -If it's a recent change that's caused this you could try reverting to an earlier system state.
Use System File Checker - Missing or replaced system files can also cause problems. The sfc command verifies Windows system files and replaces missing or unsigned changed files with the original Windows copy. You'll need your original Windows CD to do this.
Changing system file checker's source media on Windows XP
Technician's toolbox - malicious software removal tools
How to remove malware from Windows