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Microevolution, when used by creationists, refers to arbitrarily small evolutionary changes. This use posits that some kinds of evolution can happen, but the larger changes—referred to as macroevolution—cannot happen naturally. This is the evolutionary equivalent of saying that the mechanism you use to move from your bedroom to the kitchen (i.e., walking) is insufficient to get you from Boston to Los Angeles. In another sense it is like saying that watching objects fall to Earth is "microgravity" but the sun exerting influence on the planets is "macrogravity" (They might actually have it ready when needed). In short, it is a nonsensical distinction, but a useful concept for creationists to use where the evidence for evolution is impossible to deny.
When used by scientists, microevolution is the term for the gradual adaptation and evolution by a species which does not result in multiple new species. Superficially, this is the same as the definition as used by creationists, but in the scientific definitions the two concepts of micro and macro evolution are linked.
 Microevolution, according to the creationists
Microevolution is a concept generally used by creationists to denote evolution, or adaptation, within a species. This is because such adaptations can be extremely obvious and are easily demonstrated - the selective breeding of dogs being a prime example. These adaptations are only arbitrarily small, and a more concrete distinction is lacking. Some attempts have been made to say what does and does not constitute microevolution, but such things are counter-productive to creationists as it prevents them from moving the goalposts afterwards. After all, a very good tactic when faced with new evolutionary evidence is to dismiss it as "microevolution" - and if the distinction is codified, such an act becomes increasingly difficult. As such, the definition of micro and macro evolution floats and changes, and in some cases, even the development of a new species (which has been traditionally the cut-off point) has been shoehorned into the microevolution side.
In one sense, creationists can be considered to lack faith, and thus feel the need to go out and prove their assertions, rather than rely on faith. Science provides an answer, but must operate within the framework of empirical reality, and thus a conflict can arise. Science does not and cannot support the notions of Noah's Ark, as it is unfeasible. And science also can conclusively demonstrate selection pressures and their effects on the mutability of living organisms. As a result, creationists can take some science, provide an arbitrary cut-off (in this case, the micro/macro dichotomy) and still maintain their Young-Earth and special creation beliefs.
The superficially scientific discipline of baraminology makes use of microevolution in its operation. Baraminology proposes a vaguely more rigourous definition of the word "kind" that appears in the Bible in order to bring the number of animals on Noah's Ark to a more reasonable (if not much more plausible) level. It then proposes that microevolution occurred within these kinds (termed "baramins") to produce the wide variety of species seen on the planet today. For example, the "horse kind" produced everything from shire horses to zebras by partially naturalistic means.
The use of microevolution in baraminology is perhaps chosen in the mistaken belief that if science attacks it, it will effectively be attacking the whole concept of evolution. This isn't the case, however. The rate of evolution proposed by baraminology is orders of magnitude greater than any rate proposed by science, even for periods of great species diversification such as the Cambrian Explosion. Their only justification for this position is wishful thinking and as yet, in spite of their best efforts, very few people are willing to allow wishful thinking to be considered scientific evidence.
 Possible mechanisms
Explaining a distinction between microevolution and macroevolution is difficult, and creationists often ignore this entirely. One proposed explanation - indeed, the only way a complete distinction could be made - is that there would be two types of DNA. One type of DNA would be susceptible to mutation, while the other type would not, preventing a species turning into another. Needless to say, this hasn't been observed in reality.
The destruction of information, another common Young-Earth creationist gambit, could also be said to prevent macroevolution. Creationists claim that mutations can only destroy information, and thus microevolution is the result of information loss while macroevolution would be the result of information gain. Creationists propose that information gain can only be instigated by a creator, but this has also not been demonstrated in reality. There is no evidence that mutations are exclusively destructive.
 Macroevolution vs. microevolution
Put simply, macroevolution is what results when microevolution occurs in two or more populations within a species. Or maybe one might think of macroevolution as the net sum of all the microevolution that occurs within a species. Either way, the unavoidable conclusion is that it's impossible for one to be true without the other!
Perhaps an example would help clear things up. Suppose there was a large population of Species A (let's just suppose, for fun, that they're salamanders) that lives in a hilly terrain. Everything's fine and dandy until some event (like an ice age) comes along, and isolates different groups on the mountaintops. Hopefully, that makes sense so far.
It shouldn't be much of a jump to assume that those different populations, no matter how similar, still have slightly different mutations and develop under slightly different conditions. Given enough time (or enough of a difference in conditions) these two groups will develop into very different populations, and when they come back together, they may not be able to inter-breed. Et voila! New species.
So there we have it: "microevolution" is what happened to each individual group. "Macroevolution" is the result when the two groups develop under different conditions. And it may sound like an esoteric example, but guess what?! It's actually happened! The "theoretical" situation accounts for the diversity of salamanders in Appalachia.
Is it possible to have microevolution without macroevolution occuring? Sure – if there's no reproductive isolation, the species will keep micro-evolving quite merrily, no matter how long it has. But as soon as something isolates two or more groups, it sows the seeds for new species via macroevolution. And is it possible have macro without the micro? Absolutely, and there are documented cases (see macroevolution)!
 See also