What do you call this kind of entrance with a roof?

Solution 1:

In this instance, a Brit would probably describe it as a porch (the word has a slightly different connotation in American English).

Something more substantial like the one below, though, would be called a gatehouse.

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Solution 2:

In the U.S. this would probably be called a portico. The American Heritage Dictionary defines it as:

A porch or walkway with a roof supported by columns, often leading to the entrance of a building.

Where these entrances are surrounded by plantings, they are often called arbors.

Solution 3:

Maybe there's a special name that I'm unaware of, but I'd simply call that a "sheltered entrance", or an "entrance shelter".

That seems to be a widely recognized term for such a structure.