magnetic dipole moment
The magnetic dipole moment is a single number-with-a-direction that captures how strong a magnet is and which way it points. A bar magnet, a compass needle, a current loop, even a spinning electron all behave like little 'magnetic arrows'; the dipole moment is the length and direction of that arrow. It answers, how do we summarise the strength and orientation of any small magnet in one quantity?
For a flat loop of wire carrying current I and enclosing an area A, the magnetic dipole moment has magnitude mu = I A, and if the loop has N turns, mu = N I A. Its direction is along the axis of the loop, perpendicular to its plane, given by the right-hand rule (curl your fingers along the current, your thumb points along mu). This one vector controls how the magnet behaves in an external field B: it feels a turning torque of magnitude mu B sin(theta) that tries to line it up with the field, and it stores an orientation energy U = -mu B cos(theta), lowest when mu points along B.
The magnetic dipole moment is the natural language for magnetism at every scale. A compass needle has a moment that the Earth's field tugs into alignment; atoms and electrons carry tiny intrinsic moments that, added up, make materials magnetic; and it explains why a small magnet in a nonuniform field is pulled toward the region of stronger field. It is the magnetic cousin of the electric dipole moment.
A single circular loop of area 0.01 m^2 carrying 5 A has a moment mu = I A = 0.05 A·m^2. Wind it into 100 turns and the moment jumps to mu = N I A = 5 A·m^2, a hundred times stronger.
mu = N I A: more turns, more current, or more area all make a stronger magnet.
In a uniform field a dipole feels a torque but no net force; it only gets pulled bodily along when the field is nonuniform. This is why a fridge magnet is drawn toward, not just aligned with, a strong field region.