A few months ago I was looking for a Gmail replacement. I didn’t like how much of my information I was willingly giving to Google through Gmail. I tried a few different "free" email replacements and one of them was Proton Mail. I am glad to say that I have chosen to keep Proton Mail and am starting to switch over my accounts to my Proton mail email address. Proton Mail was founded in 2013 by scientists who all met at CERN.
I will keep this one short and sweet so let's get into the features of Proton Mail.
Location:
The company is physically located in Switzerland and all the e-mail servers are in Switzerland. If you know anything about Switzerland then it should be they are the most secure country to have your data (or money) stored. They have insanely strict privacy laws and will not play nice with countries trying to get your data (or money). Only a Swiss Federal Court or Cantonal Court of Geneva can compel Proton Mail to release any information they have on you. (which is not a lot, see below).
Various Account Types:
I do not pay for my account and I think the average e-mail would never either but if you need more e-mail addresses or storage space they do over very well priced plans. There is also a discount if you pay annually or for 2 years at a time.

End-to-End Encryption:
Even Proton Mail can't read your e-mails. What is in your e-mails is your information and yours alone.
Anonymous E-mail:
You do not need any personal information to create an account. Proton Mail does not track your information so you are safe.
Open Source:
Proton mail takes privacy seriously so their code is always available to the public.
Security:
Their main data-center is literally underneath 1,000 meters of granite rock under ground. It can survive a nuclear attack. Your e-mails are End-to-End Encrypted and the physical hard drives are encrypted as well. You can also have self destructing e-mails, so they get deleted after a certain amount of time.
When you log into your e-mail account you are connected to the servers in Switzerland by SSL connection. So it is double protected. Oh and the SSL Certificate Authority is QuoVadis (whom is also in Switzerland).

If you are looking for an e-mail client that has some serious security then give Proton Mail a try. The free account is definitely more restrictive than a free gmail account but for me I do not send over 150 e-mails a day so I will never have a problem just using the free account. And honestly the Plus account which is a ton more options is only $48 a year which is a steal.