Euclid's second definition is as spare as his first: a line is breadthless length. The third says that the ends of a line are points. The fourth defines a straight line as one that lies evenly with the points on itself, a phrase that has occupied commentators for two thousand years and that nobody has improved on without quietly changing the subject.
Two of the five postulates are about lines, and they are permissions rather than descriptions. The first grants that a straight line may be drawn from any point to any point. The second grants that a finite straight line may be produced continuously in a straight line. Together they say that the straightedge exists and that nothing stops it. Note what they do not say: they do not say a line is infinite. Euclid works with finite segments that can always be extended, which is a weaker and more careful claim than infinity, and the difference matters for how his geometry behaves.
The fifth postulate, the parallel postulate, is also about lines, and it is the one that would not sit still. Attempts to derive it from the other four ran for two millennia and all failed. In the nineteenth century Lobachevsky, Bolyai and Gauss independently found that consistent geometries exist in which it is false. The line is therefore the element of Euclid's system through which the system was eventually shown not to be the only one available.
Dimensionally the line is the first extension. The point has no dimension; the line has one, so it admits direction, order and distance along itself, but nothing across. Everything in plane geometry that is not a point or a curve is built from segments of it.
Euclid's line, incidentally, need not be straight. His second definition covers curves as well, and straightness is specified separately.