Question about Lee's Introduction to Topological Manifolds From page 2 in Lee's Introduction to topological manifolds: 


Question 1: What does "describe parametrically" exactly mean? Is it a synonym for
  "global coordinate chart"? (that is, an atlas consisting of only one
  element, $(M,f)$ where $M$ is the entire manifold and $f$ is a homeomorphism $M \to \mathbb R^n$?)
Question 2: Can you give me an example of a $1$-manifold that does not admit a global coordinate chart? (that is, of a non-orientable curve?) (Is the answer that there cannot be such a manifold since non-orientable means that we can embed a Moebius band in it which we can't do in dimension $1$?)
Question 3: Is this definition compatible with this
  one? What's the domain of
  the map mentioned in the definition on Wolfram Alpha? Any one
  dimensional space? I doubt it since we also want $[0,1]$ to be the
  domain sometimes. Is this Wolfram entry incorrect?

Thanks for help!
 A: It is an fact that every 1-dimensional Manifold is a circle or a piece of line. For instance, in the smooth case, you can take a look on the appendix of the book of Milnor - Topology from the differentiable viewpoint. 
From this we can conclude that every 1-dimensional can be covered by one char (in the line case) and two charts (in the circle case). I dont know if the term "describe parametrically" is synonym for "global coordinate chart", but there is no problem in interpret it like this, in the case of line. Another interpretation is just that the author is showing us how a parametrization of an 1-manifold looks like. Probably the author will answer you question better than me and this is possible because he does participate on this forum.
Edit: I have changed my anwer a litlle bit, because Kevin Carlson pointed an error. (The circle dont have just one chart).
A: Question 1: A parameterization is not quite as strong as a(n inverse of a) global coordinate chart. For example, the curve in the line given by $|t|, -1< t < 1$ does not induce a coordinate chart, since it's not $1$-to-$1$. The image of this curve does, however, admit a global coordinate chart. The circle, on the other hand, does not, though it can be parameterized by $(\cos t,\sin t)$ with $0\leq t < 2\pi$.
Question 2: The classification of smooth $1$-manifolds already discussed does extend to topological $1$-manifolds. Specifically, the connected $1$-dimensional topological manifolds, up to homeomorphism:


*

*The circle $S^1$

*$\mathbb{R}$, or $(0,1)$

*The half-open interval, e.g. $[0,1)$

*The closed interval, e.g. $[0,1]$


I don't know of a detailed, published proof of this classification, but if you have access to JSTOR, here's an outline with copious hints.
Question 3: I don't think Lee means to equate curves with $1$-manifolds. One common definition is as follows:

A curve is a continuous map $f:I\to X$, where $I\subset \mathbb{R}$ is an interval and $X$ is any topological space.

The difference from $1$-manifolds is that curves may self-intersect, e.g. as the cuspidal cubic below. This is a manifold everywhere but the origin, but there it's homeomorphic to the cross.
There are also curves that aren't $1$-manifolds anywhere. Consider the space-filling curves of Hilbert and Peano sending $[0,1]$ onto $[0,1]^2$: the images of these maps gives curves, according to the above definition, that are $2$-manifolds!
The sort of curve that is a $1$-manifold is usually called an arc-the injective image of an interval. 
Wolfram's definition is strange. Since they define their context as analytic geometry, I doubt they meant the domain should be an arbitrary $1$-dimensional topological space. That would permit curves to be disconnected, too, which they normally aren't.
