Assuming that you're running Ubuntu 17.10 (please edit your questions to include the version of Ubuntu that you're running to make it easier for us to help troubleshoot your issue), the options that you're describing are for the types of display servers that Ubuntu will use when you log in.
X.Org (or just Xorg) is an older display server that recently has been replaced by Wayland--and made the default--as of Ubuntu 17.10. However, because there are still applications that depend on X.Org, the option to use this older display server is still made available to you.
In general, using Wayland (just the "log on" option you describe) should work fine.
Assuming that you're running Ubuntu 17.10 (please edit your questions to include the version of Ubuntu that you're running to make it easier for us to help troubleshoot your issue), the options that you're describing are for the types of display servers that Ubuntu will use when you log in.
X.Org (or just Xorg) is an older display server that recently has been replaced by Wayland--and made the default--as of Ubuntu 17.10. However, because there are still applications that depend on X.Org, the option to use this older display server is still made available to you.
In general, using Wayland (just the "log on" option you describe) should work fine.