Local independence of irrelevant alternatives (LIIA), also known as Independence of Smith-dominated alternatives (ISDA), is a voting system criterion defined such that its satisfaction by a voting system occurs when the selection of the winner is independent of candidates who are not within the Smith set.
A simple way to describe is that if a voting system is LIIA, then whenever you can partition the candidates into group A and group B such that each candidate in group A is preferred over each candidate in group B, you can eliminate all candidates of group B without changing the outcome of the election.
Any election method that is locally independent of irrelevant alternatives automatically satisfies the Smith criterion, and all criteria implied by it, notably the Condorcet criterion and the Mutual majority criterion.
Contrary to what its name might suggest, it doesn't have much to do with Independence of irrelevant alternatives, which is a criterion that is actually incompatible with the Condorcet criterion.
Complying methods
Schulze and Ranked Pairs are locally independent of irrelevant alternatives. Any voting system can be "forced" to be ISDA by applying the voting system to the Smith set only.
Methods failing the Smith criterion (let alone the Condorcet criterion) never satisfy local independence of irrelevant alternatives.