Parallel lines are an interesting concept, with interesting properties. Two simple criteria dictate whether two lines are parallel. The first criterion is that the two lines exist in the same plane. The second is that they never intersect. These two facts are the only two facts that are true for all parallel lines. (Keep in mind that lines do not have endpoints, so don't be misled by incomplete drawings which only depict line segments.)
In Euclidean geometry, the geometry most people are first exposed to (and the only geometry many are exposed to), lines exist within the Euclidean plane. This plane is a flat surface, like a sheet of paper or a tabletop, except that it extends in all directions, without bounds. A number of unique properties apply to parallel lines in the Euclidean universe. For starters, given a line and a point somewhere in the plane (but not on the line), one, and only one line can be drawn through that point which is parallel to the first line. For a given pair of parallel lines, the perpendicular distance the two lines is fixed. No matter where along those two lines the measurement is made, the result is the same. (If anyone ever tells you that parallel lines meet at infinity, it's a sign that they don't understand either parallel lines or infinity.) Lines that intersect a pair of parallel lines are given the fancy name of "transversals". The name indicates that this line transverses, or crosses, both lines. Transversals get a lot of attention in geometry because they intersect both parallel lines at the same angle. (This proves very useful when trying to identify congruent or similar triangles.) Lastly, in Euclidean geometry, any line that is parallel to a given line is also parallel to every other line which is parallel to that line.
There are other geometries besides the flattened realm of Euclid, and parallel lines behave differently in each. Take the case of spherical geometry, for instance. The "plane" in spherical geometry is the surface of a sphere (like a globe). Lines are defined as "great circles", which most people will recognize as a circumference of the sphere. If the Earth were a true sphere, the equator and all the lines of longitude would be examples of great circles. The fun thing about parallel lines in spherical geometry is that they don't exist. It is easy to find a pair of lines that exist in the same plane, but they always intersect, and not just once, but twice. If this is hard to believe, it is easy to prove to yourself. Take a ball and a bunch of rubber bands. Try to find any way to put a pair of rubber bands on the ball so that each goes around its middle, but they don't overlap. You'll soon find that it can't be done. The best you can do is to put a pair side by side, but then only one is truly around the middle, while the other is slightly off center and doesn't count as a great circle.
Hyperbolic geometry is harder to visualize, but only because it is less familiar. Think of a traffic cone or a cheap paper party hat - either one is basically a hollow cone. Now imagine a second one, flipped up-side-down and stacked on top of the first. The outer surface of that pair of cones is a good image of the hyperbolic plane, except, of course, that the cones keep going and getting bigger without bounds. To make life easy, it is common to work with only half of the hyperbolic plane - the bottom cone, for instance. A line in the hyperbolic plane can be thought of as a slice through the cone. This tends to annoy people early on, because these "lines" appear curved, but that is only because we are so used to thinking in terms of the Euclidean plane. There are a number of visual aids that help when working with the hyperbolic plane. One of the most common is the Poincare Disc model, which flattens the cone down into a flat circle. Lines then appear as circular arcs, which still means that "straight" has little meaning, but at least it can be drawn on paper. Once you have a working visualization of the hyperbolic universe, you're ready to tackle parallel lines. They do exist in the hyperbolic plane. It should be quite apparent that for any slice through the cone, you can make another slice through the cone at any other point and have no overlap between those two slices. Since each slice is a line, and they don't intersect, those lines are parallel. (Grab some traffic cones and a samurai sword if you are uncertain about this.) The fun part is that by slightly changing the angle of the cut, you can make a different slice through that same point, and still not intersect the first line. In fact, for any line in the hyperbolic plane, and a point not on the line, there are an infinite number of lines which pass through that point and are parallel to the first line. Obviously, those lines intersect one another, so these parallel lines behave differently than they did in Euclidean geometry. The distance between two parallel lines is not constant. Transversals do not have to intersect at the same angle (although they sometimes can). A line which is parallel to a given line is not necessarily parallel to other lines which are parallel to that line, as you have already discovered with the infinite number of intersecting lines, all parallel to a common line.
With these three geometries, three distinct possibilities exist. Given a line and a point not on the line, one of three things happen. One parallel line exists through that point, no parallel lines exist through that point, or an infinite number of parallel lines exist through that point. This is the distinguishing factor between the three geometries examined here, demonstrating that not only are parallel lines important - they actually define the shape of the universe.