undecidable
A problem is undecidable when no algorithm can solve it correctly for every input. More precisely, a language L is undecidable if there is no Turing machine that always halts and outputs yes exactly for the strings in L and no for the strings not in L. The key words are ALWAYS HALTS and FOR EVERY INPUT: a method that works on many cases but can stall or err on some is not a decider. Undecidability is a permanent, mathematical verdict, not a comment on today's hardware or our current cleverness.
It helps to contrast undecidable with two milder situations. A decidable (also called recursive) problem has an algorithm that always halts with the right answer — this is the comfortable world we usually assume code lives in. A recognizable (recursively enumerable) but undecidable problem, like A_TM, has a machine that halts-and-accepts on the yes-instances but may loop forever on the no-instances; it is half a decider, reliable only for confirming yes. Undecidable means no full decider exists, even though a recognizer sometimes does; unrecognizable, harder still, means not even a recognizer exists.
The crucial honesty point is that undecidable is not the same as hard or slow. An undecidable problem is not one that takes a long time; it is one that no finite procedure can ever settle in general, with unlimited time and memory granted. This is a different axis from complexity entirely: complexity asks how expensive a solvable problem is, while decidability asks whether a problem is solvable at all. The halting problem, A_TM, program equivalence, and every non-trivial behavioural property (Rice's theorem) all live on the wrong side of this line.
Decidable: 'is this DFA's language empty?' (run a reachability check, always halts). Undecidable: 'is this Turing machine's language empty?' (Rice's theorem). Same surface question, opposite verdicts, because the machine model changed.
Undecidable: no algorithm halts with the right answer on every input — a limit of solvability, not of speed.
Undecidable is about computability, not complexity. NP-complete problems are decidable but believed slow; the halting problem is not slow, it is unsolvable in principle.