Pepper comes from the piper nigrum plant, which is now cultivated extensively throughout the tropics. It is a climbing plant, and on plantations is usually grown on other trees for support. It bears fruit in its third year and produces for six or seven years.
To obtain black pepper, the berries are gathered when they are turning red, but before they are fully ripe. They ferment for a few days in piles and then are dried in the sun. As they dry, they turn black. For white pepper, the berries are allowed to ripen further. After harvesting, they are soaked in a solution until the skin can be removed, revealing a white seed.
It is possible your plant produces pink peppercorns, in which case it is a type of South American rose plant. The berries of that plant are edible, but are technically not pepper.
Now, after all this rambling, it goes without saying that you should consult a botanist or county extension agent before you consume any part of that plant.