I will begin with the 2008 self-titled debut album by Scottish group Glasvegas.  Anyone familiar with this group will recognize the hits Geraldine and Daddy's Gone but this is one of those albums where some of the ones that weren't singles should have been.  This nears the top of the list of my favourite albums of all time!!

https://upload.wikimedia.org/wikipedia/en/1/1a/Glasvegas_eponymous.jpg

Daddy's Gone is one of the greatest (and most fun to sing along with) songs ever!
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I wanted the album track Polmont on My Mind to be a single, but it never was
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Hmm, no one has responded to this thread yet??

Allow me to be the first.

For me, Laundry Service by Shakira readily comes to mind. Released in November 2001; I purchased a copy of it about half a year later. It is a bilingual album (with songs in English and Spanish); all 13 tracks are totally worth listening to. :)

It's hard to narrow it down since there's so many good ones, but also to think of full albums vs. individual songs. Here's a few that come to mind:

Snoop Dogg's - R&G
Usher - Confessions
Common - Like Water For Chocolate

Here is Ultrabeat: The Album by a British electronic act called Ultrabeat.  It was their debut album, released in 2007, and spawned 6 chart singles in the UK, though a good majority of them were released a few years before the album.  I was so obsessed with this when it came out,  It's a continuous mix album, meaning each song blends into the next, kind of like the whole CD is one giant song.  I like that!

https://upload.wikimedia.org/wikipedia/en/6/63/Ultrabeat-TheAlbum.jpg


This was "technically" the first single from the album, called Pretty Green Eyes, released in 2003.  This was my favourite song of that year!
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This one called Better Than Life was the 3rd single, released in 2004.  While not my favourite song of that year, it came pretty close!
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Released in 2007, this is Boogie 2Nite, the only album released by a British female duo called Booty Luv.  You may laugh at the name, and you should because it is quite comical, but if you like fun, dance pop music, this is the album for you!  4 singles from it reached the UK Top 20 with a 5th one (as well as the previous 4) reaching the Dutch Top 40 and they are all (along with the album tracks) fantastic!  Plus, the ladies are quite easy on the eyes, so it helps just a bit ;)

https://m.media-amazon.com/images/I/61MloG2SEjL._AC_.jpg


Here is the first single, the title track, a UK #2 hit and the 73rd biggest single in the UK for 2006
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Here is the 4th single called Some Kinda Rush.  I was quite obsessed with this song when it came out
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Here is the debut 2008 solo album Skydivin' by British Hardcore DJ Darren Styles.  2 CDs = 30 tracks.  This is one of those albums where some of the songs that didn't become singles are just as good, if not better, than the ones that did

https://upload.wikimedia.org/wikipedia/en/c/cc/Skydivin.jpg


This one titled Blow Me Away was not a single, but it really should have been!
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This one called Heartbeatz, a collaboration with UK DJ Mark Breeze and UK singer Karen Danzig, was a single and a Top 20 hit in the UK.  This was (at the time) my favourite song of the 2000s.  I think it still might be my favourite!
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There's so many I could choose from that I would consider one of my favorites, a lot of good music was coming out between 2000-2009 within a lot of mainstream genres (rock, rap, R&B, etc) and there still is to this day if you look hard enough.

American Idiot- Green Day. Green Day's big comeback after a few years of not being in the spotlight, their place being taken by the newer pop punk acts in during this era before hand, like Sum 41 or Good Charlotte and most obviously Blink-182. This rock opera album took over the world pretty much by storm in late 2004 and probably until 2006 or 07ish or so.

Untitled - Blink-182. Another powerhouse of the early 2000s music scene, the last album before their hiatus in 2005. This album showed more of the mature side of Blink-182 compared to their lighthearted nature in their previous albums. I feel this album they were at their most creative and artistic point of their musical career.

Graduation -Kanye West (now known as Ye). Kanye's another artist out there who's earlier albums I enjoy more than than their newer projects, this album will always take me to that late 2000s scene of rap and R&B, when people we know now like Drake and Kid Cudi first started their career. Also when I personally started to really develop my own music taste, also.

The Eminem Show - Eminem. Honestly most of Eminem's discography from 2000-2004 I really enjoy, including his side projects like D-12. This album also pretty much took the world by storm during it's release, 8 Mile also premiering in theaters the same year.

One I forgot to name, Gwen Stefani's L.A.M.B. which came out around the same time as Snoop Dogg's R&G album. Also brings back fond memories of the Spring of 2005.

Gorillaz (2001)
Green Day - American Idiot (2004)
Gorillaz - Demon Days (2005)
All American Rejects - Move Along (2005)
Ben Folds - Over The Hedge soundtrack (2006)
Justin Timberlake - LoveSounds (2006)
Rihanna - Good Girl Gone Bad (2007)
Maroon 5 - It Won't Be Soon Before Long (2007)
Jordin Sparks (2007)
Katy Perry - One Of The Boys (2008)
Will.I.Am - Madagascar 2 soundtrack (2008)
Lady Gaga - The Fame (2008)
Britney Spears - Circus (2008)
P!nk - Funhouse (2008)
Black Eyed Peas - The END (2009)