Understanding the difference between denotation and connotation is important to understanding definitions and how concepts are used. Unfortunately, that is complicated by the fact that these terms can be used in two different ways: grammatical and logical. Even worse, both uses are worth keeping in mind and both uses are relevant to project of logical, critical thinking.
In grammar, a words denotation is whatever the word directly refers to, roughly equivalent to its lexical definition. Thus, the word atheist denotes a person who disbelieves in or denies the existence of gods. A words connotation refers to any subtle nuances that might or might not be intended by its use. For example, one possible connotation for the word atheist might be someone who is immoral and wicked, depending upon who is doing the speaking or listening.
Separating grammatical denotation from connotation is important because while one might assume that a words denotation is fully intended, whether a words connotations are intended is much more difficult to determine. Connotations are often emotional in nature, and thus if they are intended, it may be for the purpose of swaying a persons emotional reactions rather than the logical evaluation of an argument.
If there are misunderstandings about how a person is using a word in a particular debate, a primary source of that misunderstanding might lie in the words connotations: people might be seeing something not intended or the speaker may be intending something people dont see. In constructing your own arguments, its a good idea not merely to look at what your words denote, but also what they connote.
In logic, the uses of denotation and connotation are very different. The denotation, or extension, of a term is the list of a class of objects referred to by the word (think of it as how far does this word extend?). Thus the word planet denotes specific objects such as Venus, Earth, Jupiter, and Neptune. Whether it also denotes an object like Pluto is a matter of some debate among astronomers for reasons I will explain shortly.
The connotation, or intension, of a word is the list of attributes shared by all members of the class named by the word (think of it as by using this word, what do I intend?). Thus the word planet connotes certain characteristics which astronomers have decided differentiate certain objects from other objects like comets, stars, and asteroids. The debate over whether the word planet denotes Pluto is due to the fact that astronomers disagree on what sorts of attributes are connoted by the word planet, and hence whether Pluto has the right attributes to qualify as a planet.