The Condorcet Winner of an election is the candidate who, if they took part in a two-horse race against each other candidate, would win each of those elections. It should be obvious why this is a desirable quality- it demonstrates that that candidate is the 'best' for a sensible definition of the term.
Neither AV nor FPTP are guaranteed to select the Condorcet Winner, but AV is more likely to.
There's also a counterpart- the Condorcet Loser criterion. The Condorcet Loser is the candidate who, in a series of elections against the other candidate to win. That is, a universally unliked candidate. The Condorcet Loser can win a FPTP election, but can't win an AV election.
AV is far from perfect- there's actually an anomaly that can occur occasionally where downrating your preference for a candidate can actually help them, but a little analysis has shown that maybe 7 out of the 600-odd seats in the UK might suffer from it. What is unambiguous, is that FPTP creates far more troubling anomalies far more frequently.
Yes to AV, because it is a mathematically superior system.