The regeneration

The regeneration is a process by which the Time Lords, as the Doctor, can change of  shape, aspect and including character and personal tastes when they are too much sick or hurt. Time Lords has got a total of 12 regenerations, with a total of 13 incarnations, that is to say, that 13 different actors could play the same character.

On the previous video you can see the 10 first Doctor's regenerations, from the first to the eleventh, the regenerations that are missing are the one from the eight Doctor to the War Doctor, the one from the War Doctor to the ninth Doctor and the one from the eleventh to the twelfth Doctor. Now I'm going to explain the motives of each regenerations.
Be careful there could be Spoilers, specially in the videos and the regenerations explanations.

Doctor's Regenerations

First: From the first to the second Doctor. In the fourth episode of the serial The Tenth Planet, after the battle against Cybermen, the first Doctor falls down and dies. At that time the process didn't have name, but in the next episode he said that he had "renewed".

Second: From the second to the third Doctor. In the tenth episode of The War Game, the Doctor is forced to change his appearance by the Time Lords because of his crimes from the past, like steal the TARDIS.

Third: From the third to the fourth Doctor. In the fourth episode of the serial The Planet of Spiders, the Doctor suffered a radioactive poisoning in the cave of the Great One in the planet Metabelis 3

Fourth: From the fourth to the fifth Doctor. In the fourth episode of the serial Logopolis, the Doctor falls from the Pharos Project telescope when he is fighting against the Master, a moment before the regeneration the Doctor merges with the Watcher.

Fifth: From the fifth to the sixth Doctor. In the fourth episode of the serial The Caves of Androzani, the Doctor is poisoned by Spectrox toxaemia exposure on Androzani Minor.

Sixt: From the sixth to the seventh Doctor. In the first episode of the serial Time and the Rani, the Doctor is hurt when the Rani attacks the TARDIS.

Seventh: From the seventh to the eight Doctor. In the film Doctor Who: The Movie, the Doctor is shot by a gang member and he dies when he is being operated, a few hours later he regenerates.

Eight: From the eight Doctor to the War Doctor. In the episode The Night of the Doctor, he dies saving a pilot during the Time War, the Sisterhood gives him a potion which allows him to choose his personality and he chooses to be a warrior.

Ninth: From the War Doctor to the ninth Doctor. In the episode The Day of the Doctor, the Doctor is too old and regenerates after the Time War.
Tenth: From the ninth to the tenth Doctor. In the episode The Parting of The Ways the Doctor kisses Rose to absorb the vortex energy that is inside her and it destroys his cells.

Eleventh: From the tenth to the eleventh Doctor. In the second part of the episode The End of Time, the Doctor suffers a radioactive poisoning caused by the Immortality Gate.

Twelfth: From the eleventh to the Twelfth Doctor. In the episode The Time of the Doctor,he was trapped in the planet Trenzalone, but a lot of years later, before dying he received a pack of regenerations by the hand of the Time Lords.

Romana's Regeneration

Romana regenerates for unknown reasons in the first episode of the serial Destiny of the Daleks, her regeneration is very different from the Doctor's regeneration, because he doesn't feel very well after his regeneration, but she changes her appearance several times before choosing her final appearance.

Master's Regenerations

He has regenerated a few times, but only one has seen on TV, it was in the episode Utopia, he was in a planet in the future near the end of the universe, at this moment he was called professor Yana because he was hiding his identity, at the end of the episode he regenerated in a young man and he went to the present. In the episode Last of the Time Lords he was shot by his wife but he refuses to regenerate.

River Song's Regenerations

Her first regeneration was when she was a child, she regenerated in a New York alley in front of a vagabond, she didn't feel very well for the events that had happened before, at that moment she became a baby. It happened in the episode Day of the Moon.

In the episode Let's Kill Hitler, more or less 16 years later on her temporal line, she was shot accidentally by Hitler when he was trying to kill an alien who wanted to kill him. After that she helped the Doctor and she gave him her regeneration energy.