In logic and in some branches of semantics, connotation is more or less synonymous with intension. Connotation is often contrasted with denotation, which is more or less synonymous with extension. See these articles for further information.
In everyday usage, connotation has a different meaning. To explain this meaning, it is helpful to explicate the partial theory or meaning that it presupposes. The theory goes like this: every word or phrase has two kinds of meaning: primary, literal meanings (sometimes called denotations), and secondary meanings known as connotations. Connotations are thought to color what a word "really means" with emotion or value judgments.
For example, a stubborn person may be described as being either strong-willed or pig-headed. Although these have the same literal meaning (i.e. stubborn), strong-willed connotes admiration for someone's convictions, while pig-headed connotes frustration in dealing with someone.
Note that not all theories of linguistic meaning honor the distinction between literal meaning and connotations. (See Literal and figurative language.) Nonetheless, the distinction probably feels intuitively correct and seems useful to most native English speakers.
A desire for increased positive connotations (or fewer negative ones) is one of the main reasons for using euphemisms.
It is often useful to avoid words with strong connotations (especially disparaging ones) when striving to achieve a neutral point of view.