Backing up Kid Disaster's statement of Flogging Molly, and adding The Pogues to that, even though some of their songs can get a little annoying.

Off the top of my head my three favourite bands/artists I want to recommend (to anyone) are Cat Stevens, Blue October, and Pink Floyd.

Cat Stevens is basically pop/folk from the 70s. He's written some really famous songs that most people know, like The First Cut Is the Deepest, but some of his classics are really just beautiful.
This is a video of him performing Miles From Nowhere and then Father And Son live.
If you want to get into Cat Stevens, the best thing to do would be get his first album Tea For The Tillerman. If you want to be really inclusive you could get more albums, but after Tea For The Tillerman if you got the two collections (Gold and Greatest Hits), you'd have a pretty darn good Cat Stevens collection.

Blue October is a sort of alternative rock/punk rock. They're current, and I guess a little hard to classify. Just go with rock. They're biggest hit is the song Hate Me, which was nearing the tops of charts a couple years ago. I like Blue October because the vocals are really, for lack of better words, raw and intense, and the perfect example of that would be the song Hate Me. They're best album is Foiled, which is also they're most famous.

And back to the 70s with Pink Floyd, the most famous progressive rock band ever. Basically, their four big albums are all concept albums (to varying degrees), and I personally find them very involving and powerful. Apparently it's pot music. My personal favourite album would be The Wall (and the corresponding mind-blowing movie), but a lot of people desperately love Dark Side of the Moon.

I would like to suggest 3 of my top favorites. Some of their songs can be VERY heavy metal-ish, while others are more relaxed, one ven sounds a bit Celtic. The bands are(and I will post why I like them below and will do my best to post 1 or 2vids each) Sonata Arctica, Blind Guardian and Rhapsody. Now to the reasonings.

Sonata Arctica - They arent your typical Metal band. Their songs, if listened to in the order they are on a CD usually come out as a story of sorts, some of their songs change pace 2-3 times during the course of the song. And mostly, because I picked up their CD on a whim, and ended up going out the next day to buy ALL their CDs I could find. The two songs I will post by them are The End of This Chapter, which is basically a story of love, and the lighter of the two songs(on the Metal spectrum at least). And the second, is 8th Commandment. I havent quite deciphered the "story" in this one, but its the heavier of the two, and still a nice song.
YouTube - Sonata Arctica - The End Of This Chapter
YouTube - Sonata Arctica - 8th Commandment

Blind Guardian - Basically Fantasy Metal(or if you wish, Epic Power Metal). They have many, many variations of song. Some, from the Nightfall in Middle-Earth, are basically just things you would expect to see/hear in Lord of the Rings. While others, basically take you to a realm not our own, where mystical creatures could reside. And like Sonata Arctica, it was on a whim that I decided to try these guys. The first song, Theater of Pain, it just has that epic feel to it, and I think is kind of like a Light Metal style song. The second song, The Eldar, even though its from the Nightfall in Middle-Earth album, it shows their lighter side.
YouTube - BLIND GUARDIAN - THEATRE OF PAIN
YouTube - Blind Guardian - The Elder

Rhapsody - All their songs, every single one, is an addition onto their VERY first song, sort of. The first one set the stage, and the others are a continuation of the story of the "5 Valiant Warriors, who battle against the powers of Darkness". This, by far, is the best band Ive found that can continuously create songs that fit together so perfectly, that it seems like you are listening to a story, rather than music. The first song, Dargor Shadowlord Of The Black Mountain, is about, well, Dargor. Its a fairly upbeat song, and semi-heavy on the metal. The second song, Guardiani del Destino, is a nice song, with one of the "Celtic" feels to it. And not a SINGLE word of english in it.
YouTube - Rhapsody of Fire - Dargor, Shadowlord Of The Black Mountain (Full Version)
YouTube - Rhapsody - Guardiani del Destino

I'm really sad that Rhapsody is not in English Drac. I was really interested, but I have too much trouble focusing on vocals and thus story if I don't understand it.

Rhapsody is in English. At least 90% of their songs are. Some are in (which I belive, not entirely sure) Latin, or some other Souther European language that is not Spanish.

Ohhhh, I misunderstood then. I thought you meant that not a single word of any of their songs were English. xD Now I think I might have to go look into this group.

Headhunterz - Hardstyle, sounds slightly demonic and only works if you've got a stereo headset, otherwise it just sounds weird. "Battle 2 The End; D-Tuned; End of my Existence"
Dark Oscillators - Hardstyle, thick beats and is really catchy. "Stereophobia, Superstar Dj; Nero"
Dj Satomi - Trance, I have a thing for chipmunk voices, and some of the lyrics have a meaning, though his songs are basically just remixes "Castle in the Sky; Waves; Nuclear Sun"
Aya Hirano - Jpop, lyrics have meaning, easy to learn the songs, catchy. "God Knows; Lost My Music; Hero"
Dj Zany - Hardstyle; good beat paired with some sort of thing you don't hear often in songs. "Symphonic Feedback; Guardian Angel; Delomelancum"

Just to name a few.