There seem to be a number of common problem with the latest version of Windows Messenger.
What version of Windows are you using? XP or Vista?
The 1601 error is probably due to the Windows Installer service being stopped. Think of this as the part of the operating system that allows you to run installation programs.
neildownx previously gave you some instructions on how to re-start the service.
If you're using XP
Click Start -> Control Panel -> Performance and Maintenance -> Administrative Tools -> Services
Towards the bottom of the screen you'll see 'Windows Installer'. What does it say under the 'Status' and 'Startup type' columns?
Try starting the service. Right-click 'Windows Installer' and select 'Start'. The 'Status' column should change to 'Started'.
Try installing Messenger now. You may get a 1603 error, but that'll be a different problem.