What exactly is meant by a morality (Moral), in the technical sense of the word?

A system of norms and values that governs the right conduct of rational beings and claims to be valid for everyone.

The standard definition has three parts, and all three do work:

1. A system of norms — principles, values and rules, not a single opinion. Its subject matter is right action.
2. Addressed to rational beings — creatures that can understand a rule and follow it. This is why we don't hold a storm or a wasp to moral standards, and why moral responsibility tracks capacity for reason.
3. Claiming general validity — a morality asserts it holds for everyone, not just its adherents. "Don't murder" isn't offered as a personal quirk.

That third point is what distinguishes moral norms from taste and from mere convention. Someone who says "stealing is fine for me" hasn't stated a different morality — they have stepped outside the claim that makes it one.

Note that "a morality" in this sense is descriptive: it is the lived, often unwritten system of what a given society or culture treats as right and wrong. There can be several, and they can be criticised — which is exactly what ethics is for.

Go deeper:

• Stanford Encyclopedia: The Definition of Morality — the descriptive-versus-normative distinction, and what the claim to universal validity actually commits you to.
• Wikipedia: Morality — the broad overview, including how moral codes vary between societies.