Chrome(FREE) - One of the most used these days, has sync support, auto translate pages, etc. Each tab is a separate process which uses more memory but avoids killing your entire browsing session if one page has issues. Firefox(FREE) - Firefox addon support is the main benefit here. You can get adblockers, easy user-agent changing, and a whole slew of other options. Website compatibiltiy for certain sites tends to be a bit worse than Chrome though. Boat Browser(FREE) - Another great browser that has a large range of plugin support and gesture options.
Opera Mobile(FREE) - Another nice browser, not as commonly used anymore. Has a pretty nice UI, much different than Chrome. One thing you have to watch out for is that this browser uses it's own download manager. So if android kills the browser while it's in the background (due to memory) it will kill your download.
Launchers are replacements for your entire homescreen layout essentially.
Nova Launcher(FREE+PAID) - This is one of the most popular and fastest launchers out there. Paid version has additional features.
Google Maps(FREE) - Self explanatory. Nokia Here maps(FREE) - A nice offering by Nokia that supports offline maps. It doesn't support all phones though. Waze(FREE) - Real time traffic/incident reports by users with maps.
Media Playback/Sharing Apps:
MxPlayer(FREE+PAID) - Probably the best video player out there, supports software or hardware per video (depending on device support). Can play a lot of videos that the stock app can't. There's a free and paid version.
Amazon MP3(FREE) - Amazon MP3 store/cloud service. Pandora(FREE) - Streaming customized radio stations, one of the most popular streaming services. Spotify(FREE) - Another very popular streaming customized radio service. iHeartRadio(FREE) - Another very popular streaming customized radio service. PowerAmp(PAID) - Highly featured media player (local), recommended by a lot of power users. MxPlayer(FREE+PAID) - One of the best 3rd party video players out there, supports a ton of media codecs/formats and rendering modes (hardware/hardware+/software acceleration).
News Apps (Non RSS):
Press(PAID) - My favorite RSS aggregation app. Supports Feedly accounts also. I like the UI better than the Feedly app itself. Feedly(FREE) - Very popular RSS aggregation service that can sign in with your Google account.
Password/Secure Info Apps:
Keepass2Android(FREE) - An excellent encrypted password storage application + random password generator. Supports opening/saving local Keepass database files as well as remote, such as Google Drive/Dropbox/etc.
Remote Desktop Apps:
Airdroid(FREE) - Allows you to access your phone through any web browser on the same wifi network with no additional software needed. Very nice.
LogMeIn Ignition(PAID) - Remote desktop application. Fairly expensive, but it has a lot of features and support.
Splashtop Remote Desktop(PAID) - Much cheaper than LogMeIn (1/5 the price), but it doesn't have as many features. No direct file transfer between your phone and the PC. It works fairly well though.
Google Reader(FREE) - Google's RSS reader. Works pretty well.
gReader Pro(PAID) - 3rd party actually made this app, despite the name. It has some very nice features but is a bit expensive for a RSS reader.
Feedr(FREE+PAID) - Very good RSS reader. Supports GR sync, searching for RSS feeds, ability to have particular feeds bypass GR sync, and widgets. There's a free version as well.
Newsrob Pro(FREE+PAID) - Another highly regarded RSS reader. Pro version has no ads and additional features. There's a free version as well.
Pulse News Reader(PAID) - Very nice looking rss reader with a different, visual approach. The RSS and sync features are not as good as other apps though, but the visual appeal is quite nice.
Amazon(FREE) - Official Amazon app.
Ebay(FREE) - Official Ebay app.
Paypal(FREE) - Official Paypal app. Newest version supports NFC payments for NFC devices.
Social Networking Apps:
Facebook(FREE) - Official FB app for Android, works well enough. The LauncherPro FB widget is better than the FB app widget though.
Plume(FREE+PAID) - Has better widget options with multiple sizes. The only downside is that the refreshes seem to take a little longer than the native Twitter app.
TweetCaster(FREE+PAID) - Very nice twitter app, probably my favorite. Has a widget, supports themes and has a free and paid versions (free version is ad supported).
Poynt(FREE) - Local search for restaurants, events, bars, businesses, etc. Reviews, directions, weather, gas prices, etc.
TripIt (ads)(FREE) - Automatically aggregates your travel itinerary and syncs with your calendar.
TripIt(PAID) - Automatically aggregates your travel itinerary and syncs with your calendar.
Titanium Backup(FREE+PAID, requires root) - Must have tool imo. Allows easy backup + restore of apps and data, especially useful if you wipe and reload ROMs a lot. Paid version has more features like multiple backup copies per app, 1-click restore, dropbox support, etc.
Titanium Backup Pro Key(PAID, requires root) - Unlocks the pro features of TitaniumBackup. In order to use the paid version you have to install the free app, then buy and install the pro unlock key.
LBE Privacy Guard(FREE, requires root) - Must have tool, allows you to manually block or approve permissions for all installed apps. Protect your privacy.
Lux(PAID) - Very useful app if you do not like the way auto brightness adjusts on your screen. It allows you to link certain screen brightness values to the sensor values.
Genius Scan(PAID) - Extremely useful, builds PDFs for you by just taking pictures of magazines, etc.
Word Lens(PAID) - This app basically does live translations on top of what your camera sees. You have to pay per language pack. The downside is there are very limited language packs right now. The upside is that this works without a data connection, which is hugely important for traveling.
Beautiful Widget(PAID) - A number of nice looking widgets for stuff like clocks, weather, skins, toggling radios, etc. Offers similar functionality to some of the HTC sense skin widgets.
Cloud Print(FREE) - Print directly from your phone using Googles cloud print service.
ColorNote(FREE) - A nice memo/notetaking application.
Dropbox(FREE) - Store and retrieve files to your online dropbox account.
Mileage(FREE) - Mileage tracker.
Mint(FREE) - Securely track expenses, bills, various accounts, etc.
ES File Explorer(FREE) - Easily one of the best free file managers that has root access functionality. I like using Root Explorer better, but for free this is hard to beat.
Root Explorer(PAID, requires root) - Root access file browser to browse the entire android file system.
Speedtest(FREE) - Network speed test.
Volume+(PAID) - Great utility for applying Equalizer settings to your audio. Also supports increasing audio output volume for speaker, headset mode, and in-call volume (requires root).
Alive Video Wallpaper(FREE) - This app has various scenes that act as video wallpapers. The nice thing is that it's highly optimized not to cause additional battery drain (stops running when other apps are running fullscreen, when device is idle, etc).
Doom Live Wallpaper(FREE) - Great live wallpaper from the game Doom that also supports user interaction.
Haunted House HD(PAID) - Interactive haunted house with great graphics.
Angry Birds Free - Very fun time waster. Free version is ad supported.
Angry Birds Seasons Free Drag Racing - Highly addictive drag racing game with online scores and multiplayer.
Fruit Slice - Better version of Fruit Ninja imo and it's free. Graphics are much sharper than Fruit Ninja.
Pig Rush - One of my favorite games, extremely simple concept and fun to pick up and play.
Robotek - Great looking, fun game. The RNG can get annoying sometimes though...
RaiderZ - This is a really cool MMO (for PC) that I recently started playing. Game is free to play and the combat is really fun. You have to actually dodge and land attacks (ie. your sword has to actually connect in order to hit something, no auto targeting). Gameplay is fun, monsters are huge and highly interactive. Worth trying out.