What do you call a helmet that covers the entire face?
By helmet, I mean mostly medieval helmets worn by soldiers. Not helmet you wear when you ride a motorcycle although I doubt it would make any difference in this question. I am not talking about the specific helmet in the picture, but helmets in general.
The term "closed helmet" or "close helmet' refers in general to one that covers the face except for a small eye opening, such as the one in the picture. "Open helmet" or "open helm" is used for one that exposes much of the face. See the Wikipedia article on "Close Helmet" Links from this article describe and depict many types of helmets, open and closed, such as the open nasal helment
There were many individual terms for specific designs or types of closed helmet, each with a specific name, as the answer by Shino correctly indicates.
During medieval times, the name of the helmet worn by a solider depended entirely on the design. A quick search of the picture you have posted above suggests the name of this helmet in particular is a Visored Bascinet.
Other helmets that covered a solider's face included:
All these helmets vary by design. A google search should give you pictures of them. Outside of medieval times however, I don't think there is a specific name. Most are generally known as a helmet or a full-face helmet.
