The number of cipher texts possible for each plaintext
What exactly is $N_c$?

It is a number that may depend on a given ciphertext and states how many keys exist for every message that produce this ciphertext. The statement here is that every ciphertext can be produced from every message with an equal number of keys, i.e. if you draw keys uniformly at random all messages had the same probability of being the source.
For perfectly secret schemes with $|K|=|M|$ this will always be either 1 or 0. It could be 0 e.g. if the ciphertext is longer than anything in the message space for a OTP.