How to get authorized
Every tool that requires authorization should have an equipment label directing you to a wiki page that explains how to get authorized.
The general rule of thumb is that anything more dangerous than a soldering iron probably requires authorization. So, if you see something that fits that description and you don't see a label on it, then there probably ought to be one. Look on the wiki or ask an area host or someone that seems wise to help you figure out the situation.
(To be added, a link to the hosted areas and comprehensive equipment lists)
Some things need only an authorized member's say-so to count you as authorized; generally the simpler safer stuff.
Some things require a pretty serious and somewhat more formal process, maybe even a couple of sessions. More dangerous and complicated things include the ShopBot or the Tormach CNC.
Some things are in between, and need a particularly authorized proctor or trainer, but can be done in one session. Examples like the laser cutters or the vinyl cutter.
For more precise answers, use the search box on the wiki to look for the tool or equipment you're interested in, or look at the equipment label for a link to that information. Most wiki pages about tools or equipment will have a section for "How to get authorized" or similar language.
Relevant Links
https://wiki.pumpingstationone.org/Authorization_Policy
https://wiki.pumpingstationone.org/Authorization:_An_Overview_for_New_Members
https://wiki.pumpingstationone.org/How_Do_I_Get_Authorized%3F