Is there a word for a class of circular shapes?

Round shape.

A shape that is curved and without sharp angles.

Mathematically, you call them "closed curves" although that's not very satisfying. "Ellipse" also covers the two-dimensional shapes you're talking about. (Circle is a subset of ellipse in the same way that square is a subset of rectangle.)

If you just want a general-use word that describes circles and things that aren't perfect circles then you could use "hoop" or "disk" or "ring" or something else depending on the specifics.

This is mainly a response to Noldorin's answer; I don't have enough reputation points to comment directly.

An oval is not a type of ellipse. It's the other way around: an ellipse is a type of oval. Check a dictionary. Mine defines "oval" as "having a rounded and slightly elongated outline or shape, like that of an egg". It defines "ellipse" as "a regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (the foci) is constant".

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EDIT:

There has been some dispute in the comments about "oval" versus "ellipse", so I thought I would add some more citations. I claim that defining "oval" to be more general than "ellipse" is common and standard, while defining "ellipse" to be more general than "oval" is rare and non-standard. The definitions I quote above comes from the New Oxford American Dictionary. In addition to that...

dictionary.reference.com agrees: ellipse versus oval

Mathworld agrees: oval

thefreedictionary.com agrees: oval versus ellipse

mathforum.org agrees: "Simply, an ellipse IS an oval, but an oval may or may not be an ellipse."

answers.com agrees: "An ellipse always has two axes of reflection; an oval has one or more."

I've found a relatively few sources which define "ellipse" and "oval" to mean the same thing. I've found no sources at all which say that "ellipse" is more general than "oval".

Convex closed curves?

Starting with terms that others have defined the highest class is the superellipse which includes astroids ellipses and the squircle (a shape 'halfway' between a square and a circle) under ellipses you have circles (the case where major axises, or axi are equal.) otherwise you have the word I came up with cyclagon which describes any closed shape (something that doesn't point to infinity) with curved edges like a flower petal or a digon (I pronounce this like di∙jon.) the question does get more interesting in higher dimensions where you have ellipsoids, spheroids and cyclatopes in 3d and glomes in 4d. I used wikipedia for reference and spelling and glome comes from http://hi.gher.space/wiki where you can also find a crind.

Although oval is often used interchangeably with ellipse in common parlance and even by some matmaticians, I am glad to see that Wolfram now defines oval as that shape constructed from a large circle, a small circle and the pair of tangential arcs that connect them. This retains the etymology of the term oval which derives from the Latin word for egg. Interestingly enough not all eggs are oval; but so many are we ought to have a distinct word for it. Similar to this constructed shape, but not asymmetrically egg-ish, are the constructions draftsmen & architects use to approximate ellipses. Wikipedia now lists these also as a ovals, maintaining the distinction between them and true ellipses; I.e. rounded closed curves. I see this as unfortunate since again it creates ambiguity and we loose the distinct word for the true egg shape. Some mathematicians use the term quadrarc for these constructed rounded shapes (see N. T. Gridgeman, ìQuadrarcs, St. Peter’s, and the Colosseum, The Mathematics Teacher 63 (1970) 209-215). I should point out that although quadrarcs can approximate ellipses, they need not do so or at least no do so very closely, and a quick study of them shows that they can have an identity of their own. This raises an interesting comparison to the whole class of quadrarcs, superelipses, rounded rectangles, and stadia, all of which can be tweaked to appear more like a square/rectangle and less like a rounded shape or oval.