Domain theory is a branch of order theory that studies partially ordered sets which are called domains. Do not use this tag for questions about the domain of a function.
Domain theory arose in an attempt to create a denotational semantics for the lambda calculus, and has since then been used as a tool for denotational semantics of many different formal languages.
The objects studied in domain theory are posets, where the partial order is interpreted as specifying how much information an element contains, i.e. if $x \leq y$, then $x$ has more information than $y$. Such posets are called domains. In fact, the objects most studied in domain theory are called directed complete partial orders (dcpos). Directed sets are subsets of a domain where any two elements have an upper bound that is also an element of the subset. This means that given any two pieces of information, we can always find a piece of information that contains the other two, i.e. we can keep extending the amount of information. A dcpo is then a domain where each directed set has a least upper bound.
