It's the games way of forcing players together. Without it matches could last potentially forever. Shortly after a match starts a warning text should appear on the bottom half of your screen and notify you that you have 5 minutes to proceed to the marked area on the map and a timer will appear above your mini map. Open your map and you'll see a clearly marked white circle, that is the designated safe space. This is random every time. At certain points new text will remind you to move to the marked area. After the 5 minutes are up, a blue circle will appear on the map and slowly constrict until it envelops the white circle. When the blue circle starts constricting, the timer above your mini map will be replaced by a progress bar showing how close the blue circle is to enveloping the white one, and your rough position between the two circles. Being outside of the blue circle and in the blue zone damages you constantly. The further the match progresses, the more damage the blue zone does. After completely enveloping the white circle, the blue circle will replace the white one, and a new white circle will appear somewhere within the blue circle (always entirely within the blue circle). You will once again have a limited time before the blue circle will start constricting around the white circle, and this process will continue, and the the white circle will become smaller and smaller every time till the end when the safe area disappears completely, and the remaining players will take a shitload of damage and eventually die unless others die first. At this point players should be close enough to smell the cheetos on other players' fingers across computers.
Some tips: Sometimes it's hard to see exactly on the map if you are inside the safe zone or not and you don't want to move any closer than necessary. If you look at your mini map when you are outside of the white circle, you'll be able to see a dotted white line between you and the center of the white circle. The moment you enter the safe area, the dotted line disappears from your mini map. This way you know when you can stop crawling and pop a healing item for example.
Though it says you have 5 minutes to move to the marked area on the map (the first time), it actually means that the blue circle will start restricting after 5 minutes. It will take a while for the blue circle to reach the white circle. The closer you are to the safe area, the longer it'll take before the blue area catches up with you. The more experienced you become, the more time you can spend looting buildings and killing player before moving to the safe area.
You can heal and revive players in the blue zone.
Several times throughout the match, a red zone will appear on your map at random places. After a little while bombs will start dropping in this area. The area is pretty big, and bombs appear to drop at random. It's possible to move through this area while bombs drop. If you're unlucky, you'll get hit. Most of the time, you won't get hit by my experience. If you are inside buildings you are safe from the bombs (though I guess the wooden fuckboy shacks might not protect you). Vehicles will not protect you.
Aiming down the sights is the most accurate. Aiming over the shoulder is less accurate, but more accurate than hip firing. If you're inquiring whether to focus on third person or first person mode servers, that should up to your preferences. I imagine one mode will prove more popular over the other in the community. Time will tell.
