Short answer: The Doctor can now regenerate up to 24 times, for a total of 25 incarnations.
Long answer: Typically a Time Lord can only regenerate 12 times, for a total of 13 incarnations. However there have been a few events during the run of the 2005 series that must be taken into count specifically for the Doctor.
(1) After the Eighth Doctor (played by Paul McGann) was fatally wounded in a spaceship crash, he regenerated into a man who became known as The War Doctor (played by John Hurt). Due to his involvement and actions during the Time War between the Time Lords and the Daleks, future incarnations of the Doctor would try to forget this life out of shame. Also, since he wasn't known as "The Doctor", he doesn't affect the out-of-story numbering of the Doctors. So when the War Doctor regenerated, he became the Ninth Doctor (played by Christopher Eccleston), who was the tenth incarnation. So instead of the Ninth Doctor having four lives left, he now only had three.
(2) During the life of the Tenth Doctor (played by David Tennant), there was an incident that became known as a Metacrisis Event. During this, the Doctor almost regenerated once more, but after using the energy to heal himself from a fatal injury, he channeled the rest of the energy into a severed limb of his own that had been cut off soon after Nine had regenerated into Ten. This and other factors resulted in a clone of him being created (which currently lives in a parallel and sealed off dimension). In effect, the regeneration was avoided. However the energy for that regeneration was still expended. In a way, he wasted a life to save his life. But this left him with only a single regeneration left instead of two.
(3) Due to the previously stated events, the Eleventh Doctor (played by Matt Smith) should had been the final incarnation of the Doctor. However prior to his death the Time Lords had escaped their destruction from the Time War, however they had not yet found a safe way to re-enter the world. So they gave charge to the Doctor the task of telling them when things were in the clear so they could return without triggering a second Time War. Unfortunately the Doctor came under threat from the Daleks and was at risk of dying, which would remove the Time Lords' one major protector in the universe itself. So they elected to gift him with a full new cycle of regenerations just prior to the Doctor's death. Then, the Eleventh Doctor regenerated once more. During this regenerations into an entirely new cycle, the Dalek ship was destroyed and Eleven regenerated into Twelve (played by Peter Capaldi).
As things are known currently, this means that the Doctor has eleven of the twelve new regenerations left. Currently being on the Twelfth Doctor, which is his fourteenth incarnation, and barring any more non-Doctor incarnations or further Metacrisis-like events, he should be able to regenerate up to the 23rd Doctor, which would effectively be his 25th incarnation. At least 2 future regeneration have been accounted for: The Curator from Doctor Who: The Day of the Doctor, who is most likely a future incarnation of the Doctor who has retired to a life of being the Curator of the Under-Gallery of the National Gallery. Also, the character known as the Valeyard, from Doctor Who, who was revealed to somewhat be an actual future incarnation of the Doctor in the audio-adventure, Trial of the Valeyard.