There are piano books for every style of music and every composer and songwriter you can think of. You can find everything from alternative rock to zydeco, Abba to Frank Zappa. My own favorite is Celtic music, the music of Ireland, Scotland, and Wales. There are books for the music of each of these three lands and also several collections of Celtic music. There are also a few Celtic fake books. A fake book usually just gives you the melody and chord changes for each song. You can play the melody with your right hand and the chordal harmony with your left hand, either as block chords or arpeggios. Listed below are some books offering pieces in the Celtic musical tradition. Under each category, I have tried to list the books roughly by level of difficulty. If you've forgotten a lot of what you learned in those long ago childhood piano lessons, I suggest working through a good beginner piano instruction book first. I used Alfred's Basic Adult Piano Course: Lesson Book, Level One. This will help you review basic music theory and build up your skill by playing some simple pieces before tackling the ones in the books listed below.
The Chieftains -- The Long Black Veil: Piano/Vocal/Chords has a baker's dozen of songs from the CD of the same name. With their constant touring and collaborations with non-Celtic musicians, the Chieftains have probably done more than any other group to introduce Celtic music to a wider audience.
The Mary Black Song Book contains songs taken from several of her records. An effort has been made to keep each piano arrangement as close as possible to the recorded version, using the same keys and chord changes.
Wales: SONGS OF WALES 51 FAVORITE SONGS VOICE AND PIANO are mostly taken from an earlier classic book of the same name. This book includes love songs, battle songs, and songs about the beauty of the Welsh landscape. There are short background notes on each song.
The Celtic Fake Book (Fake Books) C Edition has melodies, lyrics, and chord names for over 400 songs. An index divides the songs by each nationality: Irish (separate sections for folk and popular songs), Scottish (a separate section for folksongs from the Hebrides), and Welsh.
If you're a beginner, it can be helpful to listen to a piece before attempting to play it. One of the nice things about the Internet is that a number of midi files of traditional music are available to listen to. If you're learning a traditional piece from one of the books above, you can just go to a search engine and type the name of the song and the word "midi" and the odds are good it's out there. You can also buy a compilation CD that has traditional music and listen to that. A few examples are Irish Album, Great Irish Tenors, 25 Irish Favorites, Sing Along an Irish Song, Scottish Tranquility: Piano And Orchestral Classics, Jo Stafford Sings Songs of Scotland, Harp Music of Wales, Voice of Wales, and Heritage of Wales Scotland Ireland. If you have a favorite Celtic song you're looking for, you can just type in the title here at Amazon music and it will give you a list of recordings that it's on. If you want to learn to play that particular song, you can type in "songbook" under search on the Amazon.com home page. On the results page, you should see a sponsored link leading to sheetmusicplus.com. This has a nice search feature where you can look for written music by instrument. For example, if you search under piano for "Loch Lomond", it currently shows 187 songbooks that contain the music for this piece. You can find almost any Celtic piece you're looking for here.