A slang term (derived from NP-Complete) for a group of problems in computer science that seem to require a fundamental breakthrough in Artificial Intelligence research to solve. A problem is called AI-complete if it seems to be so difficult that any AI program solving that problem would also be smart enough to solve every other problem in AI. That is by analogy with an NP-complete problem, which is so difficult that any fast solution to it would let us quickly solve all other NP problems.
The slang is usually used humorously, not literally. It reflects an attitude that these problems won't be solved by simple tricks, such as those used in ELIZA. Such problems include:
These problems are easy for humans to do (in fact, some are described directly in terms of imitating humans), and all, at their core, are about representing complex relationships between a large number of fuzzily-described concepts. Some systems can solve very simple restricted versions of these problems, but none can solve them in their full generality.