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I am trying to understand the concept of a particle's magnetic moment being 'positive' or 'negative'...

From what I understand, a negative magnetic moment means the particle's inherent 'spin' is pointing in the opposite direction of its magnetic moment... But what does that mean in terms of 'observable' properties?

Quantum 'spin' was discovered by Stern and Gerlach when they discovered that particles have an inherent magnetic moment, so... A particle's spin IS, first and foremost, it's magnetic moment...

So, how can a particle have a 'direction' to its magnetic moment in the first place, and how can it be opposite to its inherent angular momentum (spin)?

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Suppose your "particle" is actually a positively-charged sphere. Spin the sphere about some axis and the moving charges generate a magnetic field. You can use the Biot-Savart law to figure out that the direction of the magnetic field along the rotation axis will be parallel to the angular momentum.

Switch to a negatively-charged sphere, and you reverse the relationship between the direction of rotation and the direction of the current. For a negatively-charged rotating sphere, the magnetic field along the axis of rotation is antiparallel to the angular momentum vector. That's what we mean when we say that a particle has a negative magnetic moment.

(A "rotating charged sphere" is not a very good model for a quantum-mechanical spinor, but it works in this case.)

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