The "best-tux" service is handled by a DNS service called "lbnamed", which handles requests and returns a DNS response to a specific lab machine (e.g., tux-101.cae.wisc.edu). "Weighting" is the way we can prioritize certain machines or workloads to decide which tux is "best". The lbnamed find the most available lab machine by polling lab machines and weighing the results. The tuxes are scored at different values depending on base score, number of users logged in, how active the machine is, etc.
It is recommended that you use "best-tux.cae.wisc.edu" for remote connections (either ssh or vnc), since connecting directly to a specific machine could result in many users logged in at the same time, and the machine may not perform very well. Different lab machines may also not be available at certain times due to maintenance or because labs are being restructured.