The most word used is "porneia" (πορνεία), and according to Strong's means "illicit sexual intercourse" - particularly fornication, or sexual intercourse outside marriage. This of course is the real point here - Paul isn't slamming sexual desire and intercourse as a bad thing - look at Song of Solomon. What he is warning against is sex in the absence of the covenant and institution God anointed to accompany it.
This leads into the more general entry in Strong's - "idolatry" - because that's what it is - it's taking something without taking it how God intended it.
So to the question of "whose" morals - it's quite clearly God's morality - the only morality that matters. Hopefully, that lines up with the body of the church - but it needn't and often doesn't align with the morals of society. Romans 1 clearly condemns homosexuality despite the fact that it was rampant and accepted in the culture.
The point is, "sexual immorality" is not a relative term. A lust-ridden man can't justify his pornographic addiction because according to his terms, "sexual immorality" is "actual" sex. It's on God's terms. Always.