Well.. A sphere is a solid with only one surface. !
(Oh and the person who answered this before me didnt think about all the solids!)
Or if youre a surface with only one distinct sufaces/sides, or something that have no distinct inside and outside. Sphere have two distinct surfaces -- inside and outside, but they are, well, "joined" in a sense. What you will need is a Klein bottle. This object have only one side -- outside, inside whatever you want to call it, but there is only one surface.
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The second person did not think of all the solids either! A solid sphere does have only one surface, as do ellipsoids, tori, paraboloids, etc. And those are just the less irregular ones.
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A cone fits the description.
A Semi-Sphere :)
A face is a flat surface. The only solid with no flat surfaces is a sphere.
There is no solid that has those characteristics. To have an edge it must have another surface.
a sphere