My first go at this, be forgiving =]
Green Day is an American pop-punk band from California. Since conception in 1987, the band has re-invented itself several times over. Different people divide the band's career different ways, so my view might be a bit different from yours. I will cover each era to the best of my ability.
They started themselves up in the late 80's as "Sweet Children" and began to court the Berkeley, California punk scene. The name "Sweet Children" eventually gave way to "Green Day". The band released a bunch of songs/LP's as Sweet Children and Green Day, which were assembled into "1039/Smoothed out Slappy Hours" by Lookout! Records in '91. "1039" acted as the band's first album. Green Day released "Kerplunk", their second full album, on Lookout! Records in early 1992.
"1039" and "Kerplunk" are, in my opinion, some of the best Green Day out there. Some of my favorite tracks from this era are "Who Wrote Holden Caulfield?", "No One Knows", "1000 Hours", "Disappearing Boy", and "I Want to be Alone". This represents the first "phase" of Green Day - fast, bored, teen-angst punk with obvious radio-rock influences.
The band signed on to Reprise Records in '94. This deal led to the band's first major effort: Dookie. Dookie is one of the most popular Green Day albums; if you've heard of Green Day, it's probably because of Dookie. This album marked a more polished, radio-friendly Green Day, one fit for MTV heavy rotation and pop-station attention. [California fans were disgusted with the band's "sell-out", but that's life, right?] Green Day attended Woodstock '94, where they created a good bit of havoc with a mud fight. "Dookie" is an era all its own, in my opinion. Songs to note are "Basket Case", "F.O.D.", and "She". If you ask the radio, you'll also get "Long View" and "When I Come Around". Dookie represents the band's first taste of mainstream - quality recordings, slightly slower pace, popular themes, etc.
Next on the bill is 1995's "Insomniac". "Insomniac" was where the band began its first major style change, if you ask me. The songs became more experimental and less pop-punky. This album is my all-time favorite Green Day album. "Insomniac" deviates a bit from the success formula discovered in Dookie, specifically in its slowed-down, introspective lyrics. Hits from Insomniac are limited, but include "Brain Stew" and "Walking Contradiction". Tracks to note are "86", "Panic Song", and "Westbound Sign".
The next album in this era is 1997's "Nimrod". The band continued to change their image, growing more and more experimental. Tracks range from standard pop-punk "Nice Guys Finish Last", to an instrumental surf groove, to "Take Back", a dabble in punkcore, to the widely popular acoustic "Good Riddance". Nimrod was Green Day's testing phase, where they tried out different sounds to gauge where their efforts should head. Nimrod is one of my least favorite albums as a whole, but it still has some good tracks on it. "Good Riddance" has become something of an anthem in my little niche of society, however cliche that may be. My other favorites are "Scattered" and "Haushinka".
The last album from this era of Green Day was "Warning". Green Day had experimented in different directions with Insomniac and Nimrod, and had settled on Alt-rock for Warning. This album is almost not Green Day - there is nearly zero punk influence visible, with the possible exception of the semi-radio-friendly "Minority". "Minority" was the only memorable hit from this album. As always, I have a couple of favorite tracks ("Fashion Victim", "Castaway", "Church on Sunday"). I maintain my stance that this album is the first "non-Green Day" Green Day album. Some might say that "Warning" tanked in success... Some others might agree. Anyway, "Warning" was Green Day's first taste of fail.
The final and most current era of Green Day is their concept stage. This was a bolt from the blue - an established, somewhat flailing pop-punk band creating a ballad-laden rock opera? "American Idiot" was just that. Green Day seems to have found a new well of success with this, and all the power to them for it. Popular songs from American Idiot are "Jesus of Suburbia", "Holiday", "Saint Jimmy", "Wake Me Up When September Ends", "Boulevard of Broken Dreams", and, of course, "American Idiot". I'm pretty sure every song on this album has been played on the radio at some point. American Idiot was the success Green Day had been looking for in its past few albums. This album is a point of huge controversy amongst long-time Green Day fans; it doesn't fit in with the band's older music. I know a few Green Day veterans who have never listened to this album, to show their disapproval of the band's dynamic shift. Personally, I like it. It's not classic pop-punk, and it's not classic Green Day, but that doesn't mean it's not good. Favorites from this album include "Homecoming", "Letterbomb", and "Give Me Novocaine". BREAKING NEWS: AMERICAN IDIOT IS NOW A BROADWAY MUSICAL. GO FIGURE.
Green Day's latest effort is "21st Century Breakdown". This is another rock opera, in the vein of American Idiot but more sculpted and polished. I have listened to this album exactly once, and I fell asleep midway-through, so I can't tell you much about it, but I can tell you that I didn't like what I heard. My opinion of it has nothing to do with frustrations over Green Day's style change - I just don't like how it sounds. Popular songs "21 Guns" and "Know Your Enemy" get stuck in my head much too frequently. I am not an expert on Green Day anymore, it seems. =] I will have to give this album another chance, after reading reviews and whatnot, but I have yet to be motivated.
Well, there you have it. Erixouther's Guide to Green Day, take it or leave it.