Meaning of Dimension Is a (non-straight) curve in the xy-plane considered to be a 1-dimensional or 2-dimensional entity?

Depending on the answer, what is the term used to describe its other attribute (either its oneness, consisting in its similarity to a line, or its twoness, consisting in its need to be defined by at least two axes)?

Similarly, which of these categories does a line in the xy-plane not parallel to the x-axis or y-axis fall into?
The two notions you describe are both useful enough that folks have invented words to describe them. For a curve in the plane (like a circle, or a slanted line, or a horizontal line), we speak of the "intrinsic dimension" as 1, but the "ambient dimension" as two (because the object is sitting in a 2-dimensional plane). (We say that the dimension is 1 because in a small part of the shape, 1 number is sufficient to tell is where we are. On a circle for instance, points in a little part of the circle near "3 o'clock" can be described as being so many degrees clockwise from 3 o'clock or counterclockwise from 3 o'clock. This works well for the region from 2 to 4 o'clock, bad badly if we use the whole circle, because then the 9 o'clock point has two different descriptions: both 180 degrees CW and 180 degrees CCW). 

If we take that same circle and think of it as living in 3-dimensional space, then the ambient dimension becomes 3, but the intrinsic dimension is still 1. 

There's a third notion that sometimes arises, which is "how big must the ambient dimension be?" For a circle, we cannot really put it nicely into a line -- if we try to do so, we always find that some point on the line corresponds to 2 or more points of the circle (indeed, this usually happens for many points of the line). But in 2-dimensions, there's "plenty of room". 

In general, an $n$-dimensional (smooth) object will fit nicely in $2n$ dimensions, and with a mild restriction called "orientability", in $2n-1$ dimensions. Thus all orientable surfaces (2-dimensional smooth things) fit nicely in three dimensions. Some non-orientable surfaces need four dimensions to "fit" without self-intersections, though. For some folks, this "smallest dimension in which the thing will fit" gets called its "dimension" as well, so they say that a circle is two-dimensional.That's entirely reasonable, but it's not the common language used by those who study topology and geometry. 

The discussion I've given here assumes we're talking about "smooth" things, without self-intersections. A figure-8, for instance, doesn't quite fit this description, and a disc with a line drawn sticking out from one side, like a lollipop, also does not. Defining dimension for things like that is substantially trickier. 

I've also taken a topologist's view of dimension here; there are also approaches based more on analysis, which are the starting point for things like fractal dimensions, but they're more complicated (to me), and I know less about them, so I'll stop here.
Spatial dimensions are just axis perpendicular to each other. When in one spatial dimension, you can go from side to side but not up or down. To make a curve, you need to go up and down and side to side. Thus it requires two axis so it is a 2d object