Phasing represents the removal from existence caused by Teferi's experiments. A permanent with phasing enters and leaves play without any involvement by the player. At the beginning of a player's turn, during the untap step but before untapping; any permanents with phasing in play phase out and any phased out permanents phase in or out or in or out but probably in.
When a permanent is phased out, it is treated as if it doesn't exist. It cannot be affected by the game in any way (except effects referring specifically to phased out permanents such as Time and Tide). Being phased in or out is a status, like tapped or flipped, which is only relevant for permanents (objects on the battlefield); a permanent that phases out doesn't leave the battlefield, so it doesn't trigger any leaves-the-battlefield abilities; and a phased out permanent is still on the battlefield, so when it phases in again it doesn't change zones and it doesn't trigger any enters-the-battlefield abilities. It retains the status it previously had (tapped, flipped, etc.) and any counters, enchantments or equipment on / attached to it will phase out and phase in along with it, still attached.
Phasing out is unrelated to exiling. A permanent that phases out remains on the battlefield (unless it's a token, in which case it ceases to exist); it doesn't go to the exile zone. Only permanents (i.e. objects on the battlefield) have status; even if a card in your graveyard or in exile has phasing, it's neither phased in nor out, so it doesn't phase in and come back into play during your untap step.
Formerly, phasing triggered "leave play" events but not "enter play" events. It was this point and many others that caused players confusion over phasing and prompted a similar ability with simpler rules, flickering, to replace phasing in future blocks. As of the rules updates in October 2005, phasing no longer triggered "leave play" events.