Modifying normal doorbell to get notification
Many (most?) household doorbell systems in the US are powered by a simple transformer that converts 120V AC household electric supply to a lower voltage AC. Often 16V AC or 24V AC.

The coil and hammer form a device called a solenoid. When the coil is energized, the hammer (often also called a plunger) is forced in one direction, either up (against gravity) or against a return spring, striking one of the plates. When power is removed, gravity or the return spring pull the plunger back to the rest position, striking the other plate, forming the familiar ding-dong sound.

Power is applied or removed extremely simply -- by the momentary, normally-open pushbutton at your door. When you press the button, the solenoid is energized, and when you release it, the device returns to the normal state.

With regard to powering your Arduino, it is important to understand the difference between AC and DC power. The Arduino requires DC (direct-current) of either a regulated 5V DC into the 5V pin, or unregulated 7-12V DC into the VIN pin, or the regulated 5V DC from a USB device.

I don't know your level of electricity understanding, but there exists quite a bit of information on the difference between AC and DC power. Suffice it to say, you would need some additional circuit to rectify and filter and regulate the 16-24V AC into something between 5V and 12V DC.

Then, there is the issue of the loss of a good power tap while the button is pressed. If you measure the voltage across the contacts of the pushbutton while it is not pressed, you will likely find the full 16V AC or 24V AC, but if you measure those contacts while the button is pressed, you will read 0V AC. This means that tapping off of the pushbutton wires is not a good way to power a device that needs to remain on while reacting to the button press. Whether or not you could tap power off of the solenoid terminals is dependent on the voltage drop across the coil, and still you have the issue to convert the AC to DC at the proper voltage.

If this doorbell is near the ceiling, you will likely find the transformer mounted in the attic space above the ceiling near the doorbell (if there is attic space -- depending on your specific setup).  If there is a power outlet in this attic space, you could plug in a good quality AC-DC 5V adapter there to power the Arduino in the attic. The Arduino will need to remain electrically isolated from the doorbell circuit. To detect the pushbutton press, you could use a current transformer measuring the current flow through either of the 16V AC wires coming off of the transformer (measure with one of the the analog inputs of the Arduino), or with an Opto-isolator, with the transmitter connected to the doorbell circuit in a way that delivers a small current (10mA or so normally) to the transmitter side of the opto-isolator, and read the receiver side with a digital input pin on the Arduino.

Unfortunately, interfacing with a doorbell that uses AC power will require some good planning and evaluation of your doorbell system and how much additional circuitry you are willing to apply.

If you aren't careful, working near the transformer can be very dangerous to your life. Working with 16-24V AC will not be as dangerous to your life, but can easily damage the Arduino beyond repair in the flash of an eye. Take great care.