Are circles and lines in two-space one-dimensional? Circles and lines are normally regarded as one-dimensional objects. However, when embedded in two-space, they require two coordinates $(x,y)$ to specify a point within them. Are they still considered one-dimensional, and why? This is somewhat related to a separate but very similar question of why a sphere embedded in three-space is (to my knowledge) considered two-dimensional, even though it has a height, a width and a length. To laymen, spheres are generally considered 3D objects. This is where a potential misunderstanding may lie. Therefore, I am also curious of what difference, if any, there is between a mathematician's conception of what a "dimension" is versus a layman's? 

Roughly speaking, dimension refers to the degree of freedom of the geometric shape, regardless of the ambient space where that shape happens to embed in. So, lines and circles are one-dimensional, intuitively, since you have one degree of freedom at each point (you can only move along one direction). The sphere in three dimensions, much like the earth, is two dimensional since at each point you can move in precisely two independent directions. This should not be confused with the intrinsic geometry of the shape. Things like the intrinsic curvature of the shape have little to do with the dimension of the shape itself (though they can be loosely related to the dimension of the ambient space). For instance, a circle of large radius has smaller intrinsic curvature than a circle of smaller radius. Finally, particularities of how precisely a shape sits in the ambient space is a different story altogether. An important story, but different. Differential geometry is the area of study of these concepts where all of the above becomes rigorous with the help of proper definitions. The differences between how a mathematician interprets things and how a lay-person does only stem from syntactical inaccuracies. 
