Most of us recognize the idea of a definition in the context of a dictionary definition, that is, a short blurb that attempts to describe the meaning of a word in a) historical terms, b) grammatical function (noun, verb, etc.), c) through the use of simpler vocabulary, and often d) the use of synonyms. Using at least the first three of these methods is generally considered the gold-standard of a definition; the use of synonyms is sometimes seen as a bit of a cop out, and would not be accepted in place of method c. This limited scope of defining is often a problem in philosophy and the sciences, as the dictionary definition assumes a non-precise and culturally determined frame of reference, and often conflates subtle definitions.
One of the many tasks undertaken in the field of rhetoric is to analyse what methods are best for defining terms. Theoretically, informed debate requires well-defined terms that are understood by all participants. I will refrain from commenting on the current popular practice. The primary methods of defining a term are:
Synonyms: Sometimes simply giving a synonym is sufficient; this is particularly the case when using foreign terms or technical jargon. For example, 'febrile' can be safely defined as 'feverish', as the main use of 'febrile' is to indicate to your audience that you are speaking in formal terms (although, if you have to define your formal terms, this may indicate that you have misjudged your audience). This will not work in all cases; for example, defining 'hyperpyrexia' as 'fever' will lose important information.
Under this category might be listed Definition by Etymology, in which a word is defined by defining its root words; defining 'poltergeist' as a 'knocking spirit' or 'caconym' as a 'bad name' would be examples of an etymological definition.
Genus / Difference: This is the method preferred by logicians, and as you might expect, most people in the sciences. It consists of two parts; first, define the genus (the category that the word belongs to), and second, give the defining features that make it different from other members of that category. For example, an effective method of defining 'hyperpyrexia' is 'a type of fever, in which the body temperature to 41.5 °C (106.7 °F) or higher'. In some cases this may be a very simple definition; for example, in the definition 'a kitten is a young cat', 'cat' is the genus, and 'young' is the difference.
Definition by Example: Many concepts are quite vague, and are not easily sorted into categories and delineated along specific lines. Most of these are potentially definable by the Genus/Difference method, but it is not worth the trouble of constructing a precise category; for example, 'a pork-barrel government project' would be very hard to define precisely, but the idea can easily be communicated by giving a few examples. Other cases where a definition by example would be useful might be 'superstar', 'handsome', 'ghetto', 'noble' or 'jerk'. A Definition by Example might include lengthy explanations as to why X is Y, but not give clear-cut criteria that would apply to every case. This sort of definition is not very stringent, and should be avoided when possible. However, it is sometimes necessary, and the use of definition by example does not qualify as a logically fallacy and cannot be discarded out of hand.
Under this category might be listed Genetic or Historical Definition, in which something is defined by describing the origin of a the thing described; for example, giving a full understanding of a 'Water clock' would require a definition to include some information on their origin.
Negative Definition: A negative definition is often rhetorically useful, but is not generally used as a stand-alone definition. This is simple a statement of what X is not. It may be used when you are using a definition that goes against a common usage ("Male and female are not a biological terms...") or when someone wants to make a dramatic statement ("The Constitution is not a suicide pact"). A negative definition may also be found as part of a Genus/Difference definition ("A bachelor is a man who is not married").
Operational Definition: An operational definition is a definition of limited scope and use, created for a specific situation. It allows for smooth operation and works to simplify situations. For example, many governments and NGOs define 'youth' as anyone between 15 and 30. This is not the common usage, not the dictionary definition, nor any more clearly defined than, for example, the UN's definition of "between 15 and 24". However, in order to function effectively some clear, unambiguous definition must be made, and the optimal functional definition may vary from situation to situation. Operational definitions are also used in debates and philosophical arguments, where vague terms such as 'consciousness' and 'utility' need to made clear enough to use in logical arguments, even if this means using limited definitions.
Figurative Definition: The least defining sort of definition is one used purely for rhetorical purposes, without actually defining anything. "Religion is the opiate of the masses" takes the form of a definition, but clearly doesn't give a usable definition of the word religion.
Metaphors and
similes are generally examples of this. These sorts of statements may be useful in conveying attitude more than meaning; "marriage is a war of attrition" tells you a lot about where the speaker is coming from, but very little about marriage.
Any of these may be useful in defining a term in such a way that an audience can understand its use. When speaking to an audience who is familiar with the subject matter at hand, figurative definitions and definitions by example are often all that one needs to use. However, if one is making a technical argument or building a case related to a subject that the audience is not familiar with, genus/difference definitions become more important. However, it is also important to remember that the sort of definitions used may be related to one's culture and one's education. Many new age spiritualists will make statements that sound like definition by synonym, but are intended to be taken as figurative definitions ('love is life' being a very brief example), while an analytic philosopher may attempt to use genus/difference definitions on any thought that passes through his head.
When listening to an argument or thesis, you may be able to gain as much information about the reliability of the speaker from the type and appropriateness of definitions used as from the soundness of the arguments or the tone of their speech. This is particularly true when listening to politicians and newscasters, who often have to explain complex ideas to large audiences.