Differentiate between mains and a generator

I would use an opto-isolator between your doorbell and your Arduino. Make sure you get one that is suitable for the voltage and current your doorbell operates with. Measure those values with a multimeter if unsure. If your doorbell operates on AC mains then you might want to use a rectifier circuit as well because if you just throw away the negative half of the cycle your Arduino might not "see" that the button has been pressed.

Detecting that someone pushed the button should otherwise be fairly straight forward. Just have a look at the Arduino "button" example.

Realistically, not a lot. An electrical generator is any electro-mechanical device that converts mechanical energy (typically a spinning shaft) into electrical energy (a current). This is the exact opposite of the operation of an electric motor which converts a current (electrical energy) into mechanical energy. As such, some motors can also function as generators if the shaft is externally driven.

Generator is an umbrella term; there are various types of generators, but the only ones pertinent to your question are a dynamo and an alternator. A dynamo is a common generator used on bicycles to power lights, but it has been used for many other purposes as well. It incorporates a "commutator" which periodically switches the direction of the current flow from the rotor to the external circuitry to generate DC power.

If the commutator is removed, a dynamo is essentially an alternator producing AC power. The frequency of the resulting signal is determined by the windings of the generator and the rotational speed of the internal rotor. Technically, any generator that produces AC power is an alternator, but usually only the smaller AC generators driven by internal combustion engines (such as the one in a car) are known as such.