For the uninitiated: Chiptunes or 8bit music refers to music made (or made to sound like it was made) on old hardware, like Gameboys, NESes, C64, Amiga, and so on.

Best Chiptune Albums/EPs (Ebichuman Edition)
1. Dong - Gone Square: http://www.16dimensional.com/wp/?p=7
Hardly well known, but this is a work of art in melodic, progressive chiptunes. Still on my playlist two years after release. Nearly every track is so well composed they never grow old, and in fact grow better with time. You can get the English album / track names on the old label site ( http://yarhalla.jpn.org/yarhalla/16d/index_e.html ) (scroll down), if you can't read Japanese katakana. Best tracks:
Gerbera - http://yaruhara.moe-nifty.com/16d/16d002-02-dong-gerbera.mp3
Poinsettia - http://yaruhara.moe-nifty.com/16d/16d002-01-dong-poinsettia.mp3
Robiopsys - http://yaruhara.moe-nifty.com/16d/16d002-03-dong-robiopsys.mp3
Baby's Breath: http://yaruhara.moe-nifty.com/16d/16d002-06-dong-babys_breath.mp3
Bamboo Blossom: http://yaruhara.moe-nifty.com/16d/16d002-05-dong-bamboo_blossom.mp3
Lily of the Valley - http://yaruhara.moe-nifty.com/16d/16d002-09-dong-lily_of_the_valley.mp3
Narcissus - http://yaruhara.moe-nifty.com/16d/16d002-08-dong-narcissus.mp3

2. Trash80 - Icarus: http://trash80.net/content/71/Icarus-EP
Many people would choose this as the number 1 album, and they have a good argument. Expertly produced, full complex and catchy hooks, beautiful harmonies, danceable and memorable.

3. Wang-Zhi - Mind Artist Soundtrack: http://www.16dimensional.com/wp/?p=11
Another hidden gem. On its surface a "remake" of a soundtrack for an indie game, but taken on its own, a tasty little appetizer EP that wastes no time saying what it needs to say about videogame-style melodic chiptunes, and its only failing is that in its brevity leaves you wanting more.

4. Dong - Go Square: http://www.16dimensional.com/wp/?p=6
Nearly as good as Gone Square. The whole album is full of catchy and creative songs, but it takes a little more time to sink in than Gone Square. You can get the English album / track names on the old label site ( http://yarhalla.jpn.org/yarhalla/16d/index_e.html ) (scroll down), if you can't read Japanese katakana.

5. glomag - DaMaGe: http://www.8bitpeoples.com/discography/by/glomag
I almost listed this EP at number 7, just because of the two vocal tracks. They are an...acquired taste. To each his own, but my advice: Skip "End of a String" and "Tiere necken mit lange Stecken" and the four remaining tracks comprise one of the most creative, mature, concise statements of the essence of chiptunes.

6. Dong - Omoide Overdose: http://www.16dimensional.com/wp/?p=114
A set of beautiful, simple, quiet melodies. A nice little bedtime story in a decayed future world, like many an Aphex Twin piano lullaby. This stretches the limits of what you can rightly call "chiptunes," since it includes live instruments. But it's still included for its 8bit core sounds.

7. Nullsleep - Unconditional Acceleration: http://www.nullsleep.com/more.php
A straightfoward but really well done EP. At first it sounds like stock chiptunes, but the best moments are when the tracks leave behind the straightforward track structure and take risky gambles on unexpected variations of the main themes (as in Galaxy Tonight [DMG Version]).
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Live Sets / Podcasts:
Trash80 - Northwest Folklife 09
(Cleaned): http://drop.io/4u5i0xd or http://www.megaupload.com/?d=95IGH0UA
(Original): https://s3.amazonaws.com/testing-folklife-audio/audio_performances/recordings_72.mp3

A great set by Trash80. The original is available, but it's a poor recording. The clean version normalizes the sound to raise volume, boosts muffled frequencies, and adds *very* light compression.

True Chip Till Death - AHX Favorites Pt. 1 Podcast: http://truechiptilldeath.com/2009/08/tctd-podcast-007-ahx-favorites-pt-1/
Good mix of Amiga chiptunes.

True Chip Till Death - Doskpop Podcast: http://truechiptilldeath.com/2009/06/tctd-podcast-005/
Cheesy retro pop format chiptunes.
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Other Artists/Stuff:
Leeni: http://8bitcollective.com/members/leeni/
Sells her albums, but tons of free tracks on the linked 8bitcollective page. Chiptunes with actual factual beautiful well-produced female vocals. In particular, I recommend, in order:

Awake: http://8bitcollective.com/music/leeni/Awake/
Raw Footage (Trash80 Remix): http://8bitcollective.com/music/leeni/Raw+Footage%3A+TRASH80+Remix/
Evergreen: http://8bitcollective.com/music/leeni/evergreen/
Indestructible: http://8bitcollective.com/music/leeni/Indestructable
Underworld: http://8bitcollective.com/music/leeni/underworld/
Labyrinth: http://8bitcollective.com/music/leeni/Labyrinth/

Multifaros: http://multifaros.info.se/
Also has some commercial albums, but a large amount of full free albums. Generally good melodic chiptunes

Weezer - The 8bit Album: http://www.ptesquad.com/more/pte018.html
May have been posted here, I'll take that chance. If you understand "chiptunes" and "weezer," you're halfway there. But lots of surprising new directions, too.

Da Chip - Daft Punk Remixes: http://thelemurblog.com/2009/07/15/da-chip/
8bit remixes of Daft Punk. Not much more description necessary! (Thanks for the idea, crom)

One-offs from Other Artists
4Mat - Velocity: http://8bitcollective.com/music/4mat/Velocity/
Dexter - At Last: http://8bitcollective.com/music/Dexter/At+Last/
Derris-Kharlan - Theme for a Jump Goat: http://8bitcollective.com/music/Derris-Kharlan/Theme+for+a+Jump+Goat/
IAYD - When I Sleep, My Heart Speaks: http://8bitcollective.com/music/IAYD/When+I+Sleep%2C+My+Heart+Speaks/

8bit demos / NES albums: You can actually play these on your NES emulators. Many are doing this, but here's one example:
NES ROM Pulsewave flyers: http://www.youtube.com/user/balansamba
Original Pulsewave flyer ROMs: http://no-carrier.com/software.html
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Other Chiptune Sites:
Plopbox: http://www.plopbox.net/index.php
Focus on old-school tracker-based chiptunes in their native formats. Can be a bit frustrating to use the built-in media player and to navigate, but a lot of great Amiga, C64 and Atari chiptunes.

8bitcollective: http://8bitcollective.com/
This is probably the most well known chiptunes site. People post single tracks and there is rudimentary commenting. The whole range of musicians, from great to terrible, post here. However, advanced rating and useful search isn't really possible, so you probably will have to slog through a lot of crap before hitting something magical. One unfortunate rule of thumb is to look at the "views" number to guess what is worth listening to.

8bitpeoples: http://www.8bitpeoples.com/
This is a site that sporadically (but regularly) releases full albums. They break some high quality stuff (Trash80, glomag), but have a lot of other more typical chiptunes too. Good to keep an eye on.

True Chip Till Death: http://truechiptilldeath.com/
Especially nice chiptunes news / release blog.

Famicompo 01-05: http://midr2.under.jp/compo/top.html
NES competition held yearly. These files are created to play on NES emulated hardware. Two categories: Originals and Covers (from Aphex Twin to Johann Strauss).
Top 20 MP3-converted entries from Compo 05: http://www.megaupload.com/?d=Z4H56VKD

To play the others:
Play with NSF-capable player like Winamp: http://www.winamp.com/player
Winamp NSF Plugin: http://nosefart.sourceforge.net/#winamp
A nice Famicom skin: http://www.winamp.com/skins/details/146222

(Updated: Had to add some of those one-offs, see above).