Artifacts are the files you want the TeamCity server to store so that they can be downloaded after the build has finished. They will be downloadable from the TeamCity dashboard from each build.
For a .NET project you might choose the store the output of the compiler (i.e. .exe and .dll files), and the log files from running unit tests. You might just have a Windows Installer package (i.e. .msi).
It is completely up to you what gets stored for your specific needs. Just note that build artifacts do take up disk space on the TeamCity server, so if yours are large you'll want to configure the Build History Clean-up rules.