While the above categories may be of great help to those working in the field, the Bible gives a much simpler definition. It says God created everything after its own kind. So for example, we can have a dog kind and a cat kind. Thus, while through natural selection we can find in the dog kind coyotes, wolves, hyenas, etc., and through artificial selection with careful breeding we can have literally hundreds of breeds of dog, the dog kind can never change into a cat kind. This is precisely what we see when we observe all the living things around us.
So to give some kind of definition, we could say that a single kind is every species, past, present and future, that can trace its ancestry back to the same breeding pair* God created during the creation week.
The creationist case is that while minor differences may occur at the species level, the amount of change required to produce difference at the higher levels is such that it could never happen. For example, one could play a variety of different games with a pack of cards by varying the number in the hand and the use to which they are put (different Species), but they could never be used to play chess (different Domain or Kingdom perhaps).
The diagrams below will hopefully demonstrate this. The evolutionist believes as a result of a net increase in specified complexity, there has been an ever-increasing variety of life forms arising from a single ancestor, rather like the branches of a tree (lower diagram). The creationist believes God created all the different kinds of living thing, which since have increased in variety via natural selection like the roots of a forest of trees (upper diagram); but none developed into different kinds because cat kinds can turn into lions and tigers, but can never become dogs or monkeys. All the change we can see is the result of the shuffling or loss of pre-existing DNA information, and never the result of new information coding for faculties not previously seen in a species.
* God only created one man and one woman, but we are not told if He created only one male and female of all the other living species or more than one. If it was more than one, then we are referring to all those capable of interbreeding at the time they were created.