The Eighth incarnation of the Doctor (played by Paul McGann) began when the seventh Doctor was caught in a gang war gun battle. Shot twice, the Doctor died in a San Fransisco hospital circa 1980's and it was a few hours before he regenerated due to being under anesthesia when he died.
He is quickly caught up in a battle against his old adversary The Master, who has survived his execution on Skaro and is inhabiting the body of a human. As the Master knows he cannot survive for long in a human body so he plots to use the "Eye of Harmony" (an artificially created black hole on the Time Lords homeworld Gallifrey giving limitless energy and the power needed to time travel) to steal the Doctors remaining lives.
The Eighth Doctor only appeared on television in the one TV movie, which was an attempt at reviving the series and was backed by the US TV Network Fox. While this TV film was a good success in the UK (over 9 million viewers) due to poor scheduling and lack of advertising there were poor ratings in the US. As a result, the Fox Network did not back a series being created.
Ironically even though he only appeared once on our screens, in other media including radio, comic strips, novellas and Doctor Who Books , the eighth Doctor could be considered to be the longest serving Doctor (between 1996 and 2005). Adding all these media series together makes him the most prolific of all the Doctors to date. The BBC especially have created a number of novels featuring the Eighth Doctor (Eighth Doctor Adventures) which have proved popular with readers.
In terms of character, this Doctor is the romantic and quite human in his outlook and compassion, he actively encourages people to seize life and often gives people hints to their own futures.
The exact circumstances of the Eighth Doctor's death and subsequent regeneration into the Ninth have not yet been revealed. An off-hand remark by the Ninth Doctor in the 2005 episode "Rose" (commenting on the size of his own ears) suggests that the regeneration took place shortly before that story.
It is also implied that it was the eighth Doctor that bore witness to the destruction of Gallifrey in a Time War with the Daleks. It has been supposed by many that it was this Time War that caused the regeneration into the Ninth Doctor.
Stupid men are often capable of things the clever would not dare to contemplate...