An answer from the bottom up would be as follows. A group of muscle fibers is innervated by a single motor neuron: this is called a motor unit, as it activates together, since a unique axon carries the same signal to these muscle fibers. The actual message to contract the muscle is passed at the neuromuscular junction, the interface between the axon terminal and the sarcolemma (cell membrane of the muscle fiber). Here, the signal is passed chemically: the action potential reaching the presynaptic axon terminal causes the release of a neurotransmitter called acetylcholine (ACh) which travels across the synaptic cleft and binds to nicotinic acetylcholine receptors (nAChR), which then cause another action potential to fire postsynaptically at the motor endplate, travelling via the T-tubules to cause a synchronized release of calcium from the sarcoplasmic reticulum. The sarcoplasmic reticulum is sort of like a cobweb wrapped around the myofibrils (the muscle fibers) of a muscle cell, and that release of calcium, called a calcium spark, is what directly causes the muscle to contract.
Explaining precisely how the calcium causes the muscle to contract is more a matter of molecular biology, but the point is that this calcium spark only causes a temporary contraction. So it turns out, tracing this chain of transmission back to the presynaptic motor neuron, that action potentials need to be coming in constantly to keep the muscle contracted!
However, the signal to contract the muscle isn't initiated at the motor neuron, so let's trace it further. The signal travels through a ventral root of the spinal cord, coming down from the primary motor cortex, once the decision to move the muscle has been made (of course, I've skipped many steps here).
Your question of whether this is an on-and-off signal is quite interesting, as although we know that the continuous signal is required, this on-and-off idea could make sense at a higher level in the transmission chain.
So is there an on-and-off signal in the primary motor cortex (M1)? Georgopoulos et al. is the paper usually credited with discovering the way that M1 encodes muscle activation, and it turns out that as far as we're concerned, rate coding is used: that is to say, the more frequently a neuron in M1 fires, the more the muscle will contract (very roughly speaking).
Up until now, the answer is thus that a continuous signal is used, not an on-and-off one. Of course, the discussion could be taken even higher up the chain of transmission, and it very well could be that for instance the decision to move the muscle is an on-and-off signal, but the discussion would lengthen indefinitely then...
It seems to make sense to only send an on and an off signal as opposed to a continuous signal, as this is more economical, but the difficulty in actually implementing this should not be overlooked. Indeed, at some point, there would have to be a mechanism to convert the on-and-off signal to a continuous signal.