One thing - I know some of them have strange, messed up English names, but look past that. Anyway, here's a few bands I really recommend looking up on Youtube (most of them have official legal channels):
Radwimps
Well, I can go on about this band for hours, as my friends here know. One of the best bands, musically, I've ever heard. These guys have, over the course of six albums so far, covered almost every variety of music you can think of, all while keeping a distinct Radwimps sound, with picky guitar lines, genius lyrics, and most of all the singer's very unique high clear voice. The singer (Noda Youjirou), as it happens, is also the sole composer, writing all the music and all the lyrics. His voice isn't the strongest, I'll admit, but he knows how to use it, and he writes music for himself in that knowledge. And, best of all, I'm going to see them live in June *jumps around in excitement*.
Songs to look up: おしゃかしゃま (Oshakashama), セツナレンサ (Setsunarensa), 狭心症 (Kyoshinsho, 'Heart Attack'), 揶揄 (Yayu, 'Ridicule')... Well, there's a selection, but all the ones on Youtube are worth listening to.
ONE OK ROCK
These guys sound like a cross between about five punk-pop bands I used to listen to when I was in secondary school. They're brilliant. The singer, hilariously, was an original member of a cheesy pop boyband called Kat-tun, who are still very much at the top of the charts, but couldn't be more different to what he gave it up for. Thank goodness he did, though.
Songs to look up: アンサイズネア (Answer Is Near), 自分 Rock (Jibun, 'Self Rock'), Keep It Real.
シド(Sid)
Okay, this band are in the genre Visual Kei, which is an almost exclusively Japanese thing, and can seem a bit weird to people who haven't come across it before. Basically they have a very distinct look, which involves over-done hair, makeup and clothes. Actually these guys are pretty normal compared to some other ones I could name (cough cough Vidoll cough), and their music is really good. Also, it's worth looking up The Kiddie as well - they have a slightly different more in-your-face visual style (it's called Oshare Kei) but their music is also pretty good.
Songs to look up: 乱舞のメロディー (Ranbu no Melody), レイン (Rain). The Kiddie: Smile, Elite Star+.
Acid Black Cherry (ABC)
This is in fact less a band and more a vocalist with backing musicians. The vocalist in question, known simply as Yasu, was the singer of a very very famous and popular Visual Kei band called Janne D'Arc, and when they split up he went solo with new band members. He's one of the strongest Japanese male vocalists I've heard and he has one heck of a stage presence. The band's music is heavier, around the middle of the Visual Kei spectrum I'd say. One note: ABC rather like dressing up as their alter-egos, girls and geeks, and having very silly music videos.
Songs to look up: Black Cherry, 眠り姫 (Nemuri Hime, 'Sleeping Princess'), 20th Century Boys
Doping Panda
Stepping away from the rock for a minute, these guys are a bit different - punky electro dance pop, is how I can describe it. But they're fantastic. And mad. And they have a name which makes people on youtube find them accidentally while hoping to find a video of a panda on drugs. Not me, I'd like to point out, other people on the internet. Oh, they also write a lot of their stuff in English. Well, not perfect English, but you can forgive them for that.
Songs to look up: Beat Addiction, Miracle, Nothin'
I could come up with more, but I think that's enough to be getting on with. Oh, just one more thing - the song 'Tonbi' by Androp is absolutely divine, but unfortunately is their only notable track. Still very very much worth listening to, though. I had it on repeat for two days.