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?
On
An informal way to define the dimension of a curve is the following: "The dimension of the curve is the smallest number $d$ of free parameters $\theta_1, \theta_2, \ldots, \theta_d$ which are necessary to describe the curve in the form: $$\begin{cases}x = f(\theta_1, \theta_2, \ldots, \theta_d) \\y = g(\theta_1, \theta_2, \ldots, \theta_d) \end{cases}.$$
Examples:
A straight line $y=mx+q$ has dimension $1$, since it can be expressed as:
$$\begin{cases}x = \theta_1 \\y = m\theta_1+q \end{cases}.$$
A circle $x^2+y^2=1$ has dimension $1$, since it can be expressed as:
$$\begin{cases}x = \cos(\theta_1) \\y = \sin(\theta_1) \end{cases}.$$
A plane $Ax+By+Cz = D$ has dimension $2$, since it can be expressed as:
$$\begin{cases}x = \theta_1 \\y = \theta_2\\ z = \frac{D-A\theta_1-B\theta_2}{C} \end{cases},$$
assumed that $C \neq 0$.
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.