The above definitions, while giving good information, manage to skip the most basic definition of cladistics, which is organizing organisms by whether or not they share a common ancestor. As noted, this does not have to be done through DNA analysis. Dinosaurs can be arranged cladistically by evidence from their bones, for example. Cladistics is a fairly basic idea, and there is no way that any form of systematics that is based on evolution could not be based on cladistics.
The only hitch in cladistics from a lay person's perspective comes from the fact that because all organisms that share an ancestor are grouped together are referred to as a clade, two organisms with widely different ways of life and body forms are also grouped together. This is especially odd for people who hold on to the idea of an evolutionary ladder. where organisms are grouped by increasing sophistication. The prime example of this is the two warm blooded groups, birds and mammals, which were often grouped together at the top of the "evolutionary ladder". Cladistically, birds and mammals would both be reptiles, since the only common ancestor they share was a very early reptile. Birds bring us to another point, because birds were, Of course, the last off-shoot of what is conventionally referred to as dinosaurs. This leads cladistically-oriented biologists to have to use the somewhat clumsy term "non-avian dinosaurs" to refer to what we usually just call "dinosaurs". It may be somewhat difficult to describe birds as "dinosaurs" to some people, because of their different appearance, but it actually isn't too much different than referring to whales as mammals.
British biologist Colin Tudge has put forward the idea of using a complimentary term, "grade", to describe organisms that share certain morphological and ecological traits, even though they may not be a strict clade. Thus, all cold-blooded animals with scales would be in the grade "reptilia", even though it is not a true clade.