In the standard Big Bang theory of cosmology— a theory that explains a wide range of observations— distant objects show a shift in the observed wavelengths of features in their spectrum as a result of the expansion of the Universe. In between the time when the light is emitted by a distant object, and the time that we see that light, the Universe expands. The wavelength of the light goes up by a relative factor that has the same value as the relative expansion in the size of the Universe. This effect is called cosmological redshift. Because the Universe has always been expanding, we can use this to measure distance to an object, and to measure how far back in time we're looking (i.e. how much time it took the light to reach us). The more redshift we see, the more the Universe expanded, so thus the more time the Universe was expanding while the light was on its way to us, and thus the longer the trip to us was.
Astronomy has long had a handful of fringe scientists who argue that at least some of the redshifts we see are non-cosmological in origin. In particular, Halton Arp, most famous for a catalog of galaxies with disturbed morphologies (as a result of interactions), argues that quasars aren't really cosmologically distant objects at all, but are rather objects ejected from nearby galaxies, showing their redshifts as the result of an extreme Doppler shift due to their high ejection velocities. He based this originally on anecdotal observations of quasars with much higher redshifts seemingly correlated with much more nearby galaxies on the sky. For a long time, it was hard to test this correlation quantatively, for the selection effects were huge. By and large, we targeted interesting objects, but there were other interesting objects in the field. So, of all the quasars known, there was an observational bias that there would be more known near other objects that were observed for other reasons.
Various other things are claimed together with quasars supposedly being ejected from galaxies. In particular, there are claims of periodic redshifts— that is, that quasars are preferentially observed at certain redshifts, or at certain redshifts relative to the redshift of the galaxy that supposedly ejected them.
With the advent of large-scale sky surveys such as the Sloan Digital Sky Survey (SDSS), it has become possible to statistically test these predictions. Of course, the vast majority of astronomers haven't bothered, because we have extremely good models of quasars as cosmological objects that explain a wide range of observations about them, meaning that there's really no need to pay attention to the crank fringe asserting that there must be something wrong with the mainstream model. However, this (rational) response does feed into the natural tendency of many people to be attracted to conspiracy theories, who then assert that the "dogma" of mainstream science is "ignoring the evidence" for these decidedly non-conventional models of quasars. So, a few people have used the data in the SDSS to look for correlations of quasars and foreground galaxies, or to look for evidence of periodic redshifts in quasars. The result, of course, is that there is no evidence to support these theory, and indeed that the large statistics afforded by these surveys support the cosmological model.
In other words, if you want it summed up in fewer words: Arp is wrong. The evidence does not back up his arguments.
(He will disagree, and if you go to his website you can see the paranoia on the nicely designed, sparse front page. However, even if he is right about being ignored by the mainstream of science, that is because the mainstream of science has good reason to ignore him.)
Su Min Tang and Shuang Nan Zhang did a careful statistical analysis of SDSS data to look for the effects of periodic redshifts in quasars, and for correlations between quasars and galaxies. In other words, they took the predictions of Arp and his followers seriously, at least for purposes of performing the analysis. I've already stated the result above: no effects observed. Here is one of their "money" plots, Figure 7 from that paper:
The circles here are the data from the sky survey. The various lines are the results of simulations, with the error bars on the lines showing the scatter in the simulations. The solid line is the only one that's consistent with the data throughout the whole range. The various dashed lines are simulations that would result of quasars were ejected from galaxies at various different velocities.
Reference to Arp's work is also part of the larger net-crank alternate astronomy theory, "plasma cosmology" (and the even more cranky, if that were possible, "electric universe" notion, as well as modern day followers of Velikovsky). That this lynchpin has been completely debunked should hopefully help you conclude that plasma cosmology isn't anything that should be taken seriously. I hope to address more of that in later posts.