It seems strange that a field as dynamic and old as Astronomy should be burdened with so many sloppy definitions. It's a consequence of the rapid advancement of the field, where each new instrument yanks aside yet another curtain to reveal previously unimagined objects that challenge established models. Common terms, like 'Galaxy' and 'Planet' get coined as casual labels to describe vaguely understood lights in the sky, and then as our understanding deepens we find that the labels are no longer adequate.
Back in 2006, after astronomers got increasingly muddled trying to decide whether all the new objects being found at the remote edge of the Solar System were planets or asteroids, the International Astronomical Union (IAU) decided to create a proper scientific defintion for the word 'Planet'. As we all know, the outcome of that event was a definition that no longer included Pluto, which led to an awful lot of hysterical whining about having to re-write school textbooks.
Well now astronomers have found another one of their words which no longer works: Galaxy. As our telescopes get bigger and see further, we find more and more things out there that just don't fit with what think of as being a galaxy. Up till now, galaxies had a simple classification system, in which large collections of billions of stars would form in a limited set of arrangements: Elliptical, Spiral, Irregular. They have large clouds of molecular gas and dust, stars are born and die, and if a really large object fitted into one of these categories then it was a galaxy.
As with planets, some of the new objects we've found don't quite fit the categories. Hannie's Voorwerp, for example, is as big as a galaxy, but has no stars of its own - it only shines because it happens to be lit up by blasts of radiation from the core of a nearby galaxy. There could be (and probably are) many more similar objects that are totally invisible to us because they have nothing nearby to illuminate them. Does that count as a galaxy? And what about the increasing number of objects which are much smaller than any galaxy we know, but are too large to be clusters - do we label them as a new subset of galaxies (sub-dwarf, perhaps), or do just arbitrarily draw a line in the sand to say "Anything bigger than this is a galaxy and anything smaller is not"?
One possible solution, put forward by Beth Willman and Jay Strader of Haverford College, is quite appealing for its elegance. They suggest that "A galaxy is a gravitationally bound collection of stars whose properties cannot be explained by a combination of baryons and Newton’s laws of gravity.”
What does this mean? It refers to a bizarre phenomenon noticed a few decades back, where the motions of stars within galaxies seemed to violate Newton's law of universal gravitation. One prediction of Newton's laws is that the further an object orbits from its parent, the more slowly it has to move to stay in orbit. Thus, Mercury zips around the Sun at a speed of about 48 kilometers per second, while Neptune rolls along at a relatively sedate 5.5 kilometers per second. The stars in a spiral galaxy break this rule, however -- stars far from the core orbit much faster than they should. Based on what we can see, Newton's laws predict that every single one of those stars should fly outwards from their galaxies, rather than sticking to their neat mostly-circular orbits. This has led to a number of theories as to what could be holding them in check, including Newton's theories not working on very large scales, or their being something called Dark Matter which has mass and so increases the strength of a galaxy's gravitational field.
Now the evidence continues to build in favour of Dark Matter being a real thing, but nobody can yet say for certain that the problem has been solved, which is why the new definition doesn't mention it. It's a nice definition, and has a good chance of being put forward for formal approval by the IAU. The only downside is that some of what we currently consider to be galaxies will almost certainly be demoted, and given a new label. Fortunately, the outcry will probably be a great deal less than was the case with Pluto, because most people can't name any galaxies offhand, and so will never notice the change.