The Bradley effect is a political phenomenon where a voter or potential voter lies or misleads a pollster about who they plan to vote for, based on the race of the candidate. It gets its name from the 1982 race for Governor in California, where Tom Bradley, the African American candidate, lost the election to the white candidate, George Deukmejian, despite having a lead in various polls. The term "Bradley effect" is usually applied to situations involving an African American candidate, but the term also applies to contests involving members of other minorities, or women.
The "Bradley effect" is also often used incorrectly to mean a voter who will refuse to vote for any black candidate. That such voters exist is not in doubt. What is in doubt, and what makes up the discussion about the Bradley effect, is whether there are voters who will state that they are not racist, and support a black candidate, but in fact do not. And perhaps voters such as this do exist, but it is somewhat hard to figure out just how many of them are. Since Bradley effect voters are, by definition, liars, asking them whether they are likely to lie to a pollster brings up a bit of a problem. On the face of it, however, there seems to be some constraints on how big of a group it would be. Most people who would refuse to vote for a black or minority candidate will simply chock it up to their political differences, or may even proudly proclaim that they are racists. The subset of racists who are shy about their beliefs, and will lie to a pollster to appear to be moderate and open minded does not seem like a large group to me: perhaps 1 to 5 percent of the electorate. Which is, conveniently enough for me, about what the margin of error is on most polls. There is also a big question of when and where the Bradley effect shows up most often, and whether or not it is a thing of the past. These questions are of course especially important this year, because of the historic presence of a black candidate on the ballot.
I first came upon the term on the website fivethirtyeight.com, which conducts analysis of polling. The site's founder, Nate Silver seems to think it is a myth. He goes into some detail about the different reasons why he believes so, but it mostly has to do with cherry picking and over interpreting results. My favorite view on the matter expressed on the site is from someone commenting on a blog thread that the more Obama's fortunes raise, the bigger and more real some people believe the Bradley effect has become. From a possible 1 or 2 percent in the early primaries, it has steadily risen until it is now 10 percent and almost certainly by the time of election day will have to be an 80 point gap!
I think the real question that the Bradley effect is (a small) part of are the twin issues of the bandwagon and depth of support. It is very likely that voters may publicly support a candidate based on their charisma and appeal, but may rethink that when it is time to actually vote. If the motives and actions of undecided voters were well understood, elections would be much more predictable. I also believe that the Bradley effect, along with many other things, is something that will be settled one way or another in a few short weeks.