Bitcoin Private is a hard fork of Bitcoin intended to add privacy features to this legacy cryptocurrency asset. Interestingly, it's actually a co-fork, the other asset being ZCL. This project achieves privacy for its users by blocking access to the transaction's metadata such as the sender and recipient.
The transactions themselves are still visible on the blockchain. While a private version of Bitcoin would have possibly been useful years ago, I feel this coin is a little too late to be relevant. There are too many other privacy coins available with better options.
While it is possible that your Bitcoin Private investment could appreciate some value due partly to the low supply, and partly due to the use of powerful, stolen branding, I don't find it as useful as other privacy options.
They are greatly lacking in a number of features that others in the space are innovating on. Please read on to see a comparison of these other services and make this choice for yourself. I, however, don't see BTCP dethroning Monero or BTC any time soon.
You can buy Bitcoin Private on a few exchanges, but they are all smaller players. There's nothing wrong with buying from these sites, but their servers are not always reliable due to their small size. For this reason, you should be sure to immediately transfer any purchases to a safe wallet in your possession for the protection of your investment. The largest trading pair is for BTC as usual, but there are a few smaller pairs as well like LTC and Nano.
The best option for storing your coins is by using the official wallet downloads from the website. They have installation packages for Mac, Windows and Linux operating systems. As of yet, I don't believe they have any mobile options, but they are working on an integration with Coinomi for iOS and Android.