What is the time control for FIDE chess tournaments? According to the FIDE Handbook (Time Control), the time control is 90 minutes for the first 40 moves followed by 30 minutes for the rest of the game plus an additional 30 seconds per move. 

I heard from somebody working at the Annex chess club that there the time control there is 90 minutes base time and 30 second increments for both players. 

Can you clearly explain to me what "90 minutes for the first 40 moves followed by 30 minutes for the rest of the game plus an additional 30 seconds per move" means? 

Does it mean each player starts at 90 minutes with no increments until both players have made 20 moves, then after black's 20th move, each player gets 30 minutes with 30 second increments; or does it mean that each player gets 90 minutes with no increments and then after their side of the clock goes down to zero, instead of forfeiting the match, they get 30 minutes with 30 second increments? 

Also, in what situations is that time control followed and in what situations is the other time control followed? Is that time control for competitions that only people above a certain rating can join?
The clocks would start at 90 minutes, with 30 seconds added per move, starting at move one.  When move 40 (not 20 - we're counting full moves, not half moves) is completed, 30 minutes would be added (along with the 30 seconds that's added for every move.)

Events would ordinarily state what time control they are using.  If it's not on the list of major events (which is on the page you linked to) they don't have to use the time control specified for major events.