Functionalism and intentionalism are two ways to look at why people do things and how history comes together. Functionalism looks at things in terms of their practical effects, meaning what the outcome is and what functions it serves. Intentionalism looks at things in terms of why people do them, and try to understand people’s motivations and decisions.
For example, if you're looking at a law that was passed, a functionalist might focus on the outcome that the law creates, while an intentionalist would instead focus on why the law was passed in the first place, and how the people involved were thinking at the time.