This question has inspired many books. The smart-alecky answer that it is the traditional music of Ireland is not strictly wrong, but it requires more detail to be useful to the person approaching the fascinating world of Irish music for the first time. 

The human voice is the oldest instrument we know, so as we expect, there is a very old tradition of songs in Irish music. As far as we can tell, the oldest Irish songs were sung A Capella in an elaborately ornamented style. The voice was also used as an instrument for accompanying dancers when no instruments were available.

Parallel to this song tradition, there is Irish dance music. The basic styles are: jigs and reels. The reel is probably the older of the two, and many of the older reels in the tradition were imported from Scotland. It's a fast dance in 4/4 or 2/2 time. Jigs appear a little later, but by contrast, most jigs in the tradition are native to Ireland. The jig is a slightly slower dance in ternary time (predominantly 6/8 but also 9/8 and 12/8). Beside these two main styles, we have hornpipes, polkas, barn dances, waltzes and many others, imported from several parts of Europe. 

Each one of these acquired a distinctly Irish coloration when imported into the Irish dance repertoire. For example, an Irish reel is distinctly different from the Scottish version, even when the two are based on the same melody! In the same vein, Irish polkas have very little resemblance to the Polish ones.

Airs (usually slow) are instrumental versions of old song melodies. The playing of song airs reflects the old singing style, with free rhythm and mellismatic ornamentation, with great emphasis on emotive expression. We also need to mention Ireland's national composer, the harpist Turlough O'Carolan. He left an impressive body of work, which is highly prized by many modern players, but opinions about how Carolan's compositions fit into the Irish traditional music spectrum vary. Most consider them a distinct category, set apart from the more muscular dance music by its strong Baroque influences.

The root of Irish music instrumentation is the Uilleann Pipes , (pronounced ILL-un, or ILL-yun,) the so-called Irish bagpipes, which are, by the way, a much different instrument from their more famous cousin, the Highland bagpipes played by military bands. From the pipes, the tradition passed to the Fiddle (violin), then the Flute and whistle and the Button Accordion. Other instruments, such as the Banjo, Citern and Mandolin, were adopted later. The use of rhythmic accompaniment instruments such as guitar, piano, and drums like the Bodhran (pronounced BOW-rahn), are more recent developments.

The pipes are a peculiar musical instrument, because they emit a continuous tone; therefore, the piper must use ornamentation to make the melody stand out. This generates a rich and very specific system of ornaments: cuts, taps, rolls, cranns, etc. The piping ornaments were adapted to the other traditional instruments and more ornaments were added, and this ornamentation is a very important part of the Irish music sound.

The final component of our understanding of Irish traditional music is the repertoire. Nobody knows for sure how many tunes there are: thousands and thousands. It's not rare to meet people who can play well over a thousand tunes by heart. New tunes are constantly being written, of course, but only a relatively small number gain sufficient currency to become part of the tradition.

We know that there was a musical tradition in the early Irish court, in which the harp (strung with brass wires played with the fingernails) featured prominently. That tradition, however, was lost when the British subjugated and eliminated the old Irish aristocracy. After that, what survived was the dance music practiced by the poor.

There is not much documentation on the early beginnings of Irish dance music, but we can assume that a single player or maybe a small group of players provided music for dances which were held regularly in homes and churches. Traveling dance instructors were common, and dancing was an important part of social life in old Ireland. This older period is characterized by a diversity of regional playing styles.

During the years of 1845 to 1850, Ireland suffered through a terrible famine. At the end of it, fully one quarter of the population had died, and half had emigrated all over the world, but especially to America.

Once established in other parts of the world, this Irish population started looking back to its roots, including the traditional forms of music. The locus of social interaction had changed from homes and churches to the local pub, where musicians would meet after the work day and exchange tunes. The mixing of people from different areas of Ireland contributed to a reduction in regional differences, and led to a few well-established playing styles. Each region contributed its tunes to the formation of the large repertoire that we enjoy today. Recent trends suggest that much of the music has lost its connection to dancing and has become generally faster and flashier.

The nascent recording industry quickly grasped the commercial potential of this music, and generated a stream of 78 rpm recordings of the better players. For the first time, this music was being put down in permanent form, and the styles of playing that are still followed today were being crystallized. These recordings eventually made their way back into Ireland, and prepared the terrain for the renaissance of traditional dance music forms in the island.

With the formation of the independent Irish state in 1922, Irish music finally came into its own. It is now part of the traditions that the Irish state works very hard to preserve (with mixed results, of course).

The twentieth century brought the commercial success of Irish music worldwide. No longer something that the lower classes created for their own entertainment, Irish music is now big business. It retains, however, a hardcore group of aficionados who are passionate about keeping it pure. So the Irish musical tradition, being a living tradition, advances as it should: slowly but always. New instruments are added, new tunes composed, new venues are dreamed and become reality. One very positive development is the renaissance of Irish traditional dancing, and the willingness of many players to re-learn how to play music for dancers.

The premier outlet for Irish traditional music is the so-called session. (seisiun, in Gaelic). These are informal or semi-formal gatherings of musicians in pubs and other public places, to play the traditional repertoire. Despite the name, these are anything but jam sessions. Participants are supposed to know or learn the tunes and play melody, never improvisation. An informal system of leadership by seniority is enforced more or less rigidly. Despite the steep learning curve and the many obstacles that newcomers have to surmount in order to be able to join a session, they are still the best way to get involved and learn how to play Irish music.

O hAllmhurain, Gearoid, A Pocket History of Irish Traditional Music, Irish American Book company, 1998. ISBN: 0862785553

Foy, Barry, Field Guide to the Irish Music Session, Publishers Group West, 1999. ISBN: 1570982414

Vallely, Fintan (editor), The Companion to Irish Traditional Music, New York University Press, 1999. ISBN: 0814788025

O Canainn, Tomas, Traditional Music in Ireland, Ossian Publications. ISBN: 0946005