What's ProtonMail and Enigmail used for and how am I supposed to use them?

As a ProtonMail user, I rely on the security claims the company makes, meaning there are less eyes that could have potential access to your emails or private data. An example is that at the very least, most free email providers specifically states in the privacy agreements they may run algorithms through your email for advertisement purposes. Here's a few things that makes their security claim more credible than that of many email providers:

They do not store a recovery key to your emails.

Basically this leads me to believe that they do not store backup keys (compared to other email providers that lets your "recover passwords", meaning you are the only person with your key. This can mean many things, but worst case scenario, the government can't just get them to print out all your emails because, well, they really don't have access to your emails.

They have additional encryption for your email transmissions.

ProtonMail uses PGP encryption for all their outgoing and incoming emails, which verifies the identity of the user but does not identify the user or track the user. You can read about it at the link I provided.

Their physical location.

The third and somewhat debatable point for a service like ProtonMail is that their servers are located in Switzerland, they may make it more difficult for their physical servers to be seized or subpoenaed.

That being said, nothing is truly secure, if the business operates in any way in the U.S., long arm treaties like the MLAT can be used to cooperate with foreign governments to obtain the data. In cases of illegal activity, cached copies held at your ISP or on your local hard drive can be used as evidence, without ever having to decrypt the original documents.

I hope that helps to answer your question.