A "decision problem" is usually defined to be a subset $P$ of some given set $L$ (which I will refer to as a "language"), whose elements can be represented in some agreed way as inputs to a Turing Machine (or inputs to a partial recursive function or some other equivalent model of computation - I will stick with Turing Machines). The decision problem given by $P$ and $L$ is decidable if there is a Turing Machine that, given any element $\phi$ of $L$ on its input tape will terminate, writing $Y$ on its output tape if $\phi \in P$ and writing $N$ on its output tape otherwise.
So for example for SAT and TAUTOLOGY, the language $L$ is the set of all propositional formulas. SAT is the subset of $L$ comprising the satisfiable formulas (those that can be made true by some assignment of truth values to propositional variables). TAUTOLOGY is the subset of $L$ comprising the tautologies (those formulas that are true under any assignment). The method of truth tables shows that both SAT and TAUTOLOGY are decidable (as decision problems in the language $L$).
The phrase "complement of a decision problem" is not a standard technical term, but it has an obvious informal interpretation in terms of complementation of subsets of the language $L$. This interpretation does not assume anything about the internal structure of $L$ (other than the agreed representation of elements as $L$ as inputs to a Turing Machine).
In the case of SAT and TAUTOLOGY, the language $L$ has some internal structure that includes the negation operation: $P \mapsto \lnot P$. A formula $P$ is in the complement of SAT iff the negation ($\lnot P$) of $P$ is in TAUTOLOGY.
To say SAT is the complement of TAUTOLOGY is understandable but wrong - the complement of SAT comprises the formulas that are false under every assignment. The negation operation on formulas is a bijection between the complement of SAT and TAUTOLOGY, but TAUTOLOGY is not the complement of SAT.
