PGP ("Pretty Good Privacy", an understatement) allows you to send and receive emails with maximum privacy and security. Unlike other transport layer securities, with GMX PGP you are in charge. You encrypt and decrypt your emails yourself. Thus, your emails remain confidential, even if your recipient doesn't use transport layer security.
Each PGP user has one set of keys (big randomized data files) containing a public (accessible to all contacts) and a private key (only accessible to oneself). To encrypt an email, the recipient's public key is used. That way it can only be decrypted using his private key.
In the past, PGP was hard to use and required advanced IT knowledge. GMX allows you to use this technology more easily and adds features like the support for multiple devices or the cloud-based backup. In addition, setting up PGP only takes a few minutes.
End to end encryption requires all algorithms to run locally on your device (and not some remote server). This is handled by a browser addon called Mailvelope.
PGP offers an additional layer of security independent of usual transport layer security.
PGP usage is not limited to GMX customers. All you need is the recipient's public key.
In order to use PGP, you'll need either Mozilla Firefox or Google Chrome as a browser. Note that your contacts need to use PGP as well. This can be GMX PGP as well as any other implementation.
You will also need access to the recipient's public key. If he or she uses GMX PGP, the public key will be retrieved automatically.
PGP uses a mathematical one-way function to create the set of keys: Two very high randomly selected prime numbers are multiplied, which is no problem for any modern computer.
It is way harder to calculate the original two prime numbers from their product. This means that it is very hard to "guess" a private key. In fact, if the prime numbers are high enough, it is nearly impossible to achieve, even with modern super computers.