What happens to a pendulum when the mass of the arm is not negligible and What do you call this type of pendulum?

Answer: The combined inertia of the arm and pendulum would alter the energy characteristics of the system and throw off the timing.

Answer: If the mass of the arm is not negligible, then you can no longer assume (as in an ideal pendulum) that the entire mass is concentrated in the swinging object at the bottom. The center of mass would be higher up. Exactly how high depends on the characteristics of the pendulum; details can be calculated with integral calculus.