She first emerged on the national scene as John McCain's running-mate in the 2008 US presidential race. It quickly became apparent (to most observers) that she was severely underqualified for the potential job of President; this was one of many factors leading to McCain's loss against Obama.
Despite the severe underqualifications which became apparent during Palin's many media appearances for the 2008 presidential race, Palin remained a popular figure with many on the right – a fact which has mystified many on the left
Wikipedia has considerable detail about Palin's actions while in various positions; this is a summary-timeline.
Palin entered politics in 1992 when she was elected to the Wasilla City Council by a wide margin. She served a second term after winning by an even larger margin in 1995.
In 1996 she ran for Mayor of Wasilla on a platform of reducing wasteful spending and high taxes (she also raised issues of abortion, gun rights, and term limits) and was elected by a somewhat lower margin. She had apparently attracted the interest of the US Republican Party at this point, as the Alaska chapter took the "unprecedented" step of running advertisements for her. She ran again in 1999, and won by her widest margin yet.
In 2002, she ran for the Republican nomination for state governor, but came in second in a five-way race. In February 2003, she accepted an appointment to the Alaska Oil and Gas Conservation Commission, from which she resigned in January 2004. From 2003 to June 2005, Palin served as one of three directors of "Ted Stevens Excellence in Public Service, Inc.".
In 2006, she again ran for state governor, and this time was successful. In 2008, she ran as John McCain's Vice President. Despite McCain's loss to Obama, Palin inexplicably resigned as Governor in July 2008. The official story was that she was spending so much time defending herself from "frivolous" ethics complaints against her that she was no longer able to perform her job duties. Other explanations have emerged from the political right, such as the infamous claim that "the lower 48 needed her, and she heeded the call".
Since then, Palin has apparently devoted her public time and energy to media appearances, including a regular segment on Fox News.
Issuepedia has tentative hypotheses to explain Palin's popularity with both voters and power-players on the political right.
One obvious theory to explain her popularity – especially among male voters – is that they are "thinking with their gonads": they find her sufficiently attractive that their higher cognitive functions are effectively disabled, leading them to overlook her clear disqualifications for higher office. This explanation, while probably true as far as it goes, seems inadequate (to me, anyway --Woozle 20:42, 25 November 2010 (UTC)) -- it requires a complete inability to distinguish between physical attraction and qualification for office. While many voters may have difficulty with this, it seems a bit of a stretch to think that such an impairment is sufficiently widespread to explain her continuing popularity -- especially since many of her supporters are apparently well-educated and otherwise intelligent.
A more compelling theory is that Palin appeals to social dominators, who see her as both pleasant and controllable. This group is used to dominating other people (especially women) who give off certain social signals, and Palin pushes those buttons constantly. They feel safe putting Palin in a position of power (in much the same way they feel unsafe doing so with people like, say, Obama, who have their own opinions which they are capable of defending) because Palin is constantly giving cues that she can be socially dominated (whether or not she actually can be).
To a social dominator, Palin feels like an underling -- a social inferior, a secretary, "one of the girls at the office" -- who will ultimately do what he wants when it matters. What she thinks means nothing to him (regardless of how much it concerns the rest of us) because she can be overridden if she "gets too headstrong". He sees her as controllable, a servant, a minion. Any bad ideas she might have are kind of like cute mistakes made by a puppy.
The problem with voters thinking this, of course, is that she wouldn't be their minion. (Many of them, however, may have no problem with the ethics of the people she probably would be controlled by, i.e. powerful figures on the political right.)
One theory for her continued popularity with the power elite on the political right is that she serves as a distraction, a sort of agent-provocateur to prevent purposeful or meaningful political discussion from taking place. If political discussion threatens to get too sensible, she says something that is plainly idiotic but tuned to be believable to voters on the right. This forces the more sensible people away from whatever discussions were ongoing previously so they can refute Palin's claims.
They continue to find ways to fund and promote Palin's appearances because she would be a controllable minion in office and serves as a useful nuisance -- a political media troll -- in the meantime.