Nomination usually. Other mods/advisors/admins will nominate posters in the mod/advisor forums and then a period of discussion follows. Theyre looking for posters who are kind of like advisors already on topics (mine was postbac, and then added also into a preallo advisor)- posters who are a great resource for the community and can help benefit the community even more as an official advisor. After signing on as an advisor my inbox went from like 50msgs to a current 653 lol.
As for mods, I believe now, with the advisorship added into SDN, theyll look for advisors to fill the roles of mods. Theres a hierarchy
Advisors--> mods--> supermods --> admin
Just post normally as you do and if you are helpful with the majority of what you post, you will be noticed. Also note: if a particular forum has an already dedicated bunch of advisors/mods, it may be more difficult to become an advisor for that particular forum because there already exists numerous advisors.