Low frequency resonance is more often apparent in industry than high frequency resonance.  Distinction should be made between the two since the suitable strategies to solve them differ.
Mechanical resonance is a frequent and serious problem in servo systems. Control laws are often applicable for coupled loads that are assumed to be rigid. However, in practice machines consist of compliant parts. Especially parts of transmission systems in machines are compliant.  This compliance reduces stability margins, hence gains are forced to be lower and performance is reduced.
High-frequency resonance: causes instability around the natural frequency of the system.
Low-frequency:  occurs around the first phase crossover.
It is well known that for higher servo performance higher gains are desired. For stability reasons these gains are limited for compliant coupled load to motor. This compliance is inevitable and often lower than desirable due to limitations in design (lightweight/volume).
If the coupling would be rigid, the system transfer function would be an ideal integrator, however due to the compliance an anti-resonance (at F_AR) and resonance (at F_R) occurs.
In case of low-frequency resonance the problem is higher gains above F_R. The system behaves like an integrator before F_AR and like an integrator above F_R however the gain is increased. The gain of the plant becomes only the inertia of the motor, the load is effectively disconnected.  The gain must often be lowered which results in poorer command and disturbance response.
PM: difference of -180 and the phase of the open loop at 0 dB gain. 
GM: is the negative of the gain of the open loop at -180 degree phase.
High-frequency resonance occurs if the first natural frequency lies very well above F_R in a only lightly damped mechanism. The gain near F_R results in a strong peak. It is caused by high stiffness transmission in combination with low damping.  This is often the case in high-end servo machines. Whilst smaller and more cost-sensitive machines typically have less stiffness in that case low-frequency resonance occurs.