A DFA is often defined as a restricted type of NFA. If $\Sigma$ is the input alphabet and $Q$ is the set of states, the transition structure of an NFA is specified as either a relation $\rho \subseteq Q \times \Sigma \times Q$, or as a function $\delta : (Q \times \Sigma) \to 2^Q$. If we adopt the latter definition, then we can say that an NFA is deterministic if $|\delta(q,\sigma)| \leq 1$ for all $q\in Q$ and $\sigma \in \Sigma$, and complete if $\delta(q,\sigma) \neq \emptyset$, again, for all $q \in Q$ and $\sigma \in \Sigma$.
A word is accepted by an NFA if it has an accepting run. A deterministic automaton has at most one run. A complete automaton has at least one run.
Some authors define trim automata as those in which each state is on some path from an initial state to a final state. For certain languages, you cannot have automata that are both trim and complete. In those cases, it is convenient to keep the completeness requirement out of the definition of deterministic automaton.