I'll mention the informal definition of point and line from the work of Euclid :Euclid's Elements.
A point is that of which there is no part.
And a line is a length without breadth.
And a surface is that which has length and breadth only.
consider a straight rectangular rod of length $L$ width $W$ and height $H$.
Now to measure the space allocated by this rod you will need to consider its volume given by $L\times W \times H$ . the objects which we come across in day to day life have these three parameters which define the space allocated by them completely. now consider a thin $A4$ size paper of a very fine width as shown. we are not very much interested in its width because its width doesn't allocates much space . we are usually interested in the area allocated by a paper. In mathematics we have such a paper of width $0$ . This $0$ width entity is called a surface in mathematics like gaussian surface. These are called two dimensional . Similarly we have a line in mathematics which have $0$ width and $0$ height but have measurable length. These lines are called one dimensional.
now we come across something called an arc in mathematics which is an extention of a non-straight line but still an arc has no width and no height. The mathematical entity arc can be described by only one coordinate as explained in anna v's answer, does the dimensionality of space in which the entity is described determines the dimensionality of the entity itself NO! the arc is still one dimensional, it is the space in which it is plotted three dimensional or two dimensional, it has still only its length called the arc-length as its unique description. you may signify any number of coordinates to any point allocated on the arc to describe its position in space.
e.g consider a straight line in three different situations
Irrespective of the coordinate system the line will remain one-dimensional.
Similarly your loop like mathematical strings are one dimensional whether you place it in a three or four dimensional space it doesn't matter.
An algebraic plane curve given by one polynomial equation $ƒ(x,y) = 0$ is one dimensional.
e.g a circle given by $$ƒ(x,y) = x^2 + y^2 - 1 = 0$$ is one dimensional.
reference : http://en.wikipedia.org/wiki/Plane_curve