The purpose of a definition paragraph is to fully and clearly explain a term or a subject, and this is generally done using a combination of three different methods: providing a synonym, placing the topic within a larger category, or providing an explanation of what the topic is NOT in order to better explain what it IS.
Topic Sentence
The topic sentence (be definition) identifies the topic of your paragraph, and in a definition paragraph you may choose to do that by providing a dictionary definition. This will get the ball rolling and you can highlight specific elements of the definition, or characteristics of the topic in each of your supporting sentences.
Appropriate signal words, like transitional phrases, can help your reader to understand the meaning and purpose of your paragraph. In a definition paragraph use:
is defined as... refers to...
is called... is described as...
means... a term / concept...
Synonym
Synonyms are words that have the same or very similar meanings. When you write a definition paragraph using synonyms, you are using words that have meanings very close to the meaning of your topic in order to make your reader understand your topic. You might choose words or concepts that are simpler and more easily understood to illustrate the meaning of a more complicated topic. For example, take the concept of hyperbole.
Most people know what it means to exaggerate, even if they don't know what hyperbole is. If you define the concept of hyperbole by using it's synonym exaggeration it will be easier for your reader to understand the new information because you've associated it with something they are already familiar with.
Class
As in, a group of objects or concepts sharing common characteristics, not that place you go to get writing assignments. When you define something by identifying the class that it belongs to, you are essentially identifying it's most foundational characteristic and then defining it according to that characteristic. For example, what is a mango? A fruit. What is curling? A sport. What is a silverfish? A highly revolting and disturbing bug. All of these concepts have other characteristics as well - a mango has a red skin with patches of yellow and/or green and is a light orange on the inside, while a silverfish has so many legs that it looks like eyelashes - but these one word definitions can be applied at the most basic level. Once your topic has been identified by class you can start applying some of those other characteristics to differentiate it from other related concepts and give your reader a more complete definition of your specific concept.
Negation
When you define by class, you tell your reader what your topic is at it's most basic level, but when you define by negation you do the opposite. Take the silverfish. (Seriously. Take it. Take it far away.) A silverfish is neither a fish, nor is it necessarily silver. What about a peanut? Neither a pea, nor a nut. Especially in situations where a topic has a confusing or misleading name, establishing what it definitively is NOT is a good first step to explaining what it is.