Why doesn't my iPhone 6 ring?

The phone will not ring if Do Not Disturb is enabled, either manually or on a schedule that you provide in the Settings app. When it is enabled you will see a crescent moon in the top status bar. That's one reason the phone won't ring.

Another is if you have assigned a silent ringtone to the contact that is calling you.

It's also possible that the call isn't reaching your phone from your carrier's network. In this case you will not see the calling number in the Recents list in the phone app. This can happen if the signal is weak, if you have changed locations recently and the network hasn't found you yet, or, if you are on a CDMA network, you are actively using cellular data (CDMA carriers in the US cannot handle simultaneous voice and data). This last situation is getting rarer in the US, as both CDMA carriers (Sprint and Verizon) are upgrading their networks to LTE, but some partner carriers and some exurban towers have not yet been upgraded.

Thank you! Thank you! Thank you!

I've been missing calls all week and I could not figure out why! Important calls - like from doctors and cable companies and so on.

I couldn't figure out what was going on.

Then I saw your comment and then I saw the crescent moon on my iPhone.

I can't really figure out how it got set that way. I don't specifically remember doing that though I think I did play around in this area a couple of weeks ago. Can't imagine I was missing calls for weeks though.

Anyways. Thank you so much. Big problem solved!