String vibration - what is Tension really measuring? The equation $v = \sqrt{T/\mu}$ is a way of calculating wave-speed using tension and linear string density.

However, usually tension is described to be actually the weight of the load rather the tension acting against the load (upwards from the load rather than downwards in the picture).
Is it because $F=ma$ and $ma = L({\rm load}) - T$, since accelerating of the system is $0$ then $T=L$? Or is it something else I'm misunderstanding...?
 A: Tension is simply the force that is transmitted through the string. If you were to attach a string to the ceiling, and then attach a mass to the bottom of it, then in order for the system to be in equilibrium the resultant force on both the string and the mass would have to be zero.
If the resultant force on the mass is to be zero then the string must exert a force on the mass, equal in size to the weight of the object, but acting upwards.
By Newton's Third law, if the string exerts an upwards force on the mass, then the mass exerts a downwards (equal and opposite) force on the string, equal in magnitude to the weight of the mass.
Therefore (and finally) for the string to have no resultant force acting on it, the ceiling must exert an upwards force on the string, equal in magnitude to the weight of the mass.
And therefore again by Newton's Third law, the string must exert a downwards force on the ceiling (We do not consider the other forces acting on the ceiling, but just say that it is fixed as this logic could go on forever).
The point is that the string exerts both a force on the mass and on the ceiling. Both forces act towards the middle of the string (away from the points of suspension) but are equal in magnitude.
The forces both depend on the mass of the object being suspended (this was what the magnitude of all the forces discussed above depended on).
The tension in the string is the force that is transmitted through the string that allows the system to be in equilibrium. Without the tension there would be no upwards force acting on the mass, and the mass would fall. we say that the tension acts all the way through the string.
I hope this helps.
