The form is that of an apse, though they are not usually free-standing.

it's also a wedding ceremony

Are you asking constructively (I mean in a constructive terminology)?

If the answer is yes, in spanish that kind of structure are called "semi-cupula", I don't know what would be the english term (maybe semi-spherical dome or something like that). Basically, it is an arc rotated at its geometric center around the z-axis. 

A similar type of structure (open-air, open-faced pavilion for public "performances") is the bandshell.

Vaulted half-dome.

check out Hadrian's Villa ... much sweeter

It's called a grotto.

The one in the picture seems to be constructed through corbelling - basically stacking rocks with a little bit of an an overlap. The earliest domes in the world were built this way. Otherwise Miles nailed this grotto for being a vaulted masonry half-dome. 

Actually I'm working on a project which has one of these Grotto's on site.  Very similar in scale and look.  It is on the grounds of a Maronite Catholic Church Site.  The plan is to remove the Church and save the Grotto.   Then build a new Church on the site.