Literally, a bandwagon is a wagon that carries the band in a parade. Riding on the bandwagon is popular since one can enjoy the music, conveniently without walking. The phrase "jumping on the bandwagon" is therefore used in the sense of "joining an increasingly popular trend". This is especially true for voting: Here, the bandwagon effect refers to the phenomenon that people vote for those candidates or parties at stake who are likely to succeed. By that, they hope to be on the 'winner's side' in the end.
In microeconomics, bandwagon effect is a technical term for an interaction of demand and preference. The bandwagon effect arises when people's preference for a commodity increases as the number of people buying it increases. This interaction potentially disturbs the normal results of the theory of supply and demand, which assumes that price and preference are independent. See also Veblen effect.
In science specifically, the bandwagon effect is the phenomenon of scientists exercising self-censorship when reporting results that differ considerably from "accepted wisdom". For example, when measuring important values in astronomy, such as the distance of the Sun to the center of the Milky Way, published values tend to agree with the accepted value at the time of publication, even if completely new measurement procedures are employed. When a measurement yields a result at variance with the then-accepted value, scientists will check their methods and calculations repeatedly and delay (or even abandon) publication, while results that agree with the accepted value are published uncritically.
Without the bandwagon effect, one would expect that published estimates for such a constant initially show a wide range of values and then converge over time as measurement precision increases. In reality however, published values often cluster closely together, with the whole cluster moving over time in a certain direction.  shows evidence of this effect.