Ley lines are, at the very basic end, alignments of places. These can be of either geographical, historical or mythological significance—depending very much upon which ley line theory you subscribe to. Or even whether you call them simply "leys" (which are already "lines"), as their discoverer (or inventor) did. At the very beginning of modern ley line theory, only established (physical) places like ancient monuments and megaliths, natural ridge-tops and water-fords were relevant. These were the places the amateur archaeologist Alfred Watkins connected with what he called "leys" (from 1921, in his books on "Early British Trackways" and "The Old Straight Track").
Alfred Watkins and the Discovery of Leys
The very name and our modern concept of ley lines started with Alfred Watkins. While he drew on earlier sources and read about possible astronomical alignments of ancient sites (similar to those found at, say, Newgrange or Stonehenge), his personal observations around Blackwardine in Herefordshire started in 1921 and formed the basis of his theory. They came upon him as a sort of sudden revelation, and he was skeptical at first, not quite trusting his map alone. Checking from a higher vantage point, he found that crossroads, ford, standing stones, wayside crosses, causeways, hill forts and ancient churches (most on mounds) seemed to align in a way that formed a definite track through the landscape.
The line thus created was named a "ley" by Watkins ("ley lines" is thus a redundant tautology)—as many of the lines discovered by him simply passed through places with names containing the syllable "ley" (or spelling variations of this). In his theory, "leys" were laid out by "dodmen" to help travelers traversing the (then quite forested) countryside. That some roads still ran (and, indeed, still run) on these leys was further evidence to Watkins.
It is of note that Watkins saw lays as a "road network" with signposts, nothing more. It should also be kept in mind that Watkins' leys were not super-highways from Land's End to John O'Groats, but local affairs.
His theory was, however, shot down by established archaeologists and historians—mainly on the basis that the countryside surveyed has a huge number of (possibly) relevant objects and that any grid with a generous helping of randomly placed points will have a huge number of "alignments". Basically, the argument against leys goes, it may all be down to chance. Which was "proven" by the famous "telephone leys" archaeologist Richard Atkinson "found" by joining the dots marking telephone boxes on a map.
A counter-argument may well point out that telephone boxes are generally placed next to the busiest roads, which may again be running on ancient leys ...
To the point: while Alfred Watkins' theory of leys is at the same time fascinating and frustrating, it has not been dis-proven. then again it is almost impossible to prove the non-existence of something.
New Age Revival
While Watkins' original work was no longer seriously discussed in established academic circles after a few years, a new interest in his theories came with the dawning of the Age of Aquarius. In 1969 the writer John Michell single-handedly revived "ley lines" as a study subject, now with a definite mystical and New Age twist.
Michell took Watkins' down-to-earth theory from the local to the global level, mixed in a dose of Chinese feng shui (at least as it is understood or interpreted in the West) and created a highly spiritualised version of the basic idea, which has been adopted by and expanded upon by numerous other authors and applied to both local landscapes and ever more far-reaching, continent-wide alignments. Which, upon closer and less enthusiastic scrutiny, often fall literally flat due to simple map-making or -reading problems (a globe is not flat, after all) and miss the point literally by miles (due to alignments drawing on small-scale maps between "points" the size of small countries).
While Watkins' theory cannot ultimately be dis-proven and has the physical evidence to support it, Michell's theories (and much more so those ever more exotic ones of his latter-day followers) often rely on the perceived importance of certain points and a certain belief system. From amateur archaeology and landscape observation, ley lines have progressed to an almost religious status.
Ultimately any visitor to Ireland may observe a huge number of alignments (in the local, Watkins way) - whether these are marking ancient tracks, or even more, is more than often down to what the observer wants to believe. But it is a fun way to explore the landscape—and you might never know to which attractive place the next ley may guide you.