Relevance is how well a set of returned results matches what the person searching actually wanted, as opposed to what they literally typed.
Relevance has two halves that are worth separating. Recall is whether the right item was retrieved at all; precision is whether it appeared near the top. A system can fail either way, and the fixes are different — recall problems are usually matching or indexing problems, precision problems are usually ranking problems.
It is inherently subjective and business-inflected. For an identical query, the most relevant result may be the closest textual match, the item in stock, the highest-margin option or the one this specific visitor is most likely to buy. Every production search system is a negotiation between those definitions.
Because relevance is judged by humans, it can only be improved by looking at humans: click position, refinement chains, abandonment, and zero-result logs. Tuning relevance without reading the query log is guessing with extra steps.