Technically we say the one on the left is smooth and the one on the right is singular . Formally the definition of smooth and singular depend on derivatives, and we'll look at that later in A parade of Surfaces. For now we'll try to get an intuative definition by looking at tangent planes.
First lets consider a curve.
At any point on the curve we can find a line which just touches the curve, but does not cross the curve. Such lines are said to be tangent to the curve, lines which cross the curve are called transverse . For example, tangent transverse. Any point on this curve, has a unique tangent. Occasionally you can have inflections on curves where the curve changes from turning one way to turning the other way.