When you enter the Chamber of Lament, from Everfall or from Cassardi beach portal, you will see an "accessing server" message, during which a copy of the online Ur dragon will be downloaded. Kill the 3 enemy pawns to summon the Ur dragon into the arena. The Ur dragon will come with hp (and weak spots status) representative of his global health, but at any time players may be greeted with Ur dragon of varying hp. Do not be alarmed if you see an Ur dragon with hp different from what others are reporting - he is not healing himself. Look at the mean for its general health rather than individual hp values.
You will then battle him for 8 minutes on your PS3, after which he will fly away and you will see another "accessing server" message, during which the damages you did will be uploaded to the server for consolidation into the global health.
Over time, as players contribute their effort towards the common goal, the Ur dragon's global health will decline. Eventually, when the Ur dragon is killed by anyone, we go into grace period (GP). If you enter the Chamber of Lament during grace period, you will have a chance to land your own killing blow to gain good rewards (loots, and he will dragon forge every item you equipped).
During grace period, his health will start with around 0.3 bars and will decrease over time. So you may happen to summon a 0.3, 0.2, 0.1 bars or even one with hardly any hp left. You have to fight him as usual - this is a real fight, not one that you can destroy each red spot with a hit or two. We usually call this the 'hard' grace period.
As time passes, eventually we get into a stage where he will be summoned with barely a sliver of hp and all red spots lit up. Each red spot is easily destroyed, so go destroy them all and get your rewards. We usually call this the 'easy' or 'soft' grace period.
The total grace period is less than 30 minutes, so timing is critical. You do not want to be spending so much time fighting a 'hard' grace period Ur dragon such that, should you be unable to kill it, you also miss the 'easy' grace period by then. If you are not confident in tackling the 'hard' dragon, exit the arena, go back to the inn and rest, then enter again to catch it with a lower hp or catch the 'easy' grace period. You can also use ferrystone to get to the inn quickly or even use Godsbane to just reload from last save.
Disclaimer: A lot of the above are speculation from observation. There is no confirmation from Capcom.