I suppose there aren’t any strict restrictions on what can be used as a Celtic music instrument but there are more usual, and less usual ones. Here are some of the more usual ones and maybe a mention of some others too.
The fiddle is one of the classic Celtic instruments. Appearing everywhere in all Celtic traditions. Producing a wide range of tones and expressing a wide range of feelings.
The whistle, sometimes called the penny, or tin whistle is another very common Celtic instrument. Traditionally most Irish musicians learn this, even if they play something else too. coming in many sizes from low D to high D (or even E).
The beautiful sound of the Uilleann pipes can be heard in many a Celtic session. In Norwich, on a Wednesday evening at the Gatehouse Pub for instance.
The characteristic sound of the Irish flute is more mellow and airy than the modern metal ones and more difficult to play (so I’m told).
The Guitar is now used universally in Celtic music. Often tuned with a more open tuning, such as DADGAD for a bigger open sound which suits the modal nature of many Celtic tunes.
The Bouzouki is very popular for Irish music. The modern form is usually flat backed, more like a mandola, rather than the round backed Greek variety.
The Banjo in it’s 4 string Tenor form is very popular with Irish musicians. usually used as a melody instrument rather than a chordal one and played with a plectrum.
Perhaps the mandolin is less common than some other instruments in Celtic music but is still widely used. Usually tuned the same way as the fiddle and with the same range of pitch (GDAE). The related mandola is also common, though this is miss named as it is actually an octave mandolin (i.e. tuned an octave lower than a mandolin), were as the mandola is tuned the same as a viola (CGDA).
The button accordion, often referred to as the melodeon, is widely found in Celtic music. The Irish version is usually the B/C which gives to melody player higher versatility but renders the bass end almost useless. The D/G version is perhaps less versatile but the basses can be used and still gives the player access to the most widely used key signatures.
The concertina is used quite commonly for Celtic music usually tuned less ‘wet’ than for English music.
A little controversial as a Celtic instrument but nevertheless used by quite a few Celtic bands. Lunasa being a good example