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Android is now a Mobile OS platform which cannot be neglected however skeptic you are about the whole openness of Android. It certainly has benefits over other OSs and it's business model is helping to gain and grab market share from it's competitors.
Android is the second biggest App store, second to Apple's. Though I would claim it has an app equivalent to most apps on other platforms, getting the right app can be a little cumbersome since Android does not have a very unified / organised App market. Also, it's openness has been criticized ( appreciated too by many) to attract independent developers which might lead to security loop holes.
Okay, so let me cut the crap and get to some awesome apps you should have on your Android device. I am covering only smartphone apps for now ( the Android tablet app market is still very naive to comment on), and only the ones which are not on the build already, like Maps, Gmail, etc. Also, I am refraining from mentioning about games. The apps are not in the order of quality, rather in the order of frequency of use.
Launcher Pro
Android is open, fine! But most device manufacturers misuse this. We have the Touch Sense UIs, the Motoblurs, the Touch Wiz interfaces and so on sitting on top of the default Android UI and they suck big time. Launcher pro gives you a ton of options for UI customization.
Facebook
Okay, the Facebook app for Android is far from perfect but is definitely the most used ( not useful, FB itself is such a waste of your precious productive time) app. The iOS counterpart is better in many ways but as there is not a very good alternative to this, Android users spend most of their smartphone time on this app. Certain features like sharing Facebook albums' photographs, etc are present on other platforms yet though. The Google + app is pretty cool too.
Advanced Task Killer Free
Android's true multi-tasking can be a little tricky sometimes. This gives rise to the need of manually managing the memory resources just like in our good old Windows. Of all the task managers out there, including the in-stock one, the Advanced Task Killer Free is the cheapest ( $0, can't get cheaper than that) and easiest to use.
Rock Player
This is probably the best video player out there for Android which supports most formats. The lite version has ads which do not interrupt the video substantially. The video is smooth and clear and satisfactory on a mobile phone.
Chrome to Phone
Register your phone and Chrome browser with this app and you are granted the power of sending any web link from your browser to phone instantly. It's more like Read-It-Later functionality, but on your phone.
Dropbox
This is the best productivity app out there. The option to view, download and share documents or any file for that matter makes it very appealing and a must-have. Documents are viewed using QuickOffice and the built-in pdf reader by default, which can be changed if alternate document viewers are installed.
Brightest Flashlight Free
Most Android phones have a camera and an LED flash too. The OS does not allow the flash to be continuously on and hence this app. I found this app easy to use and very simplistic. It makes a funny noise when launched though.
Google Music
Google Music Beta is an online music storing service where you can save your purchased ( or stolen for piracy sites) music in the cloud and play it from any device. There is an app for this on Android which plays your saved music both from your device and the cloud. The UI is impressive but a little laggy / buggy I must say.
Netflix
Netflix is finally out on most Android devices after having several hardware acceleration issues especially with NVIDIA processors. This is a must have for Netflix on-the-go users who watch Instant-Play videos on Netflix.
Browsers
The built-in browser is pretty slick and quick but lacks features like tabbed browsing, private browsing, etc. There are several browser replacements on the Android platform, viz. Skyfire, Firefox, Opera, Dolphin, etc, etc. Dolphin browser's interface is neat and includes tabbed browsing. InBrowser can be used for private browsing.
Pulse
Pulse is a content aggregator with a beautiful interface. Several websites, blogs, RSS feeds can be added to this app for a one-stop access. You can also log into this app using Facebook connect to see the links and videos posted in your network.
QR Droid / Red Laser
These two apps are content directors. Use your phone's camera to point to a barcode or QR code and the app will direct you to a web page, app, give you contact details, product page with pricing and reviews, etc
Team Viewer
After a long wait i got Team Viewer installed on my Atrix without having to root the device. Team Viewer is the best fre VNC client. You can control your computer, share your desktop, etc using this VNC client. Another option is LogMe In Ignition, which is not free.
HeyWire
Heywire is a text/tweet/chat client. As it's tagline says " Text, tweet and chat like crazy", it certainly allows you to do that. The latest updates made this app fast and responsive. Facebook chat and twitter are integrated right into this app. This can be used to send free SMSs. The app issues you with an alternate phone number with which you can send SMSs. Pretty useful in the US where incoming and outgoing messages are charged $0.20.
Tango
Tango allows users to make cross-platform ( Andoid to iOS and vice versaFaceTime. Other services like Fring and Skype can also be used.
Sound Hound
Sound Hound works on an algorithm which identifies a song based on a recording a part of the song and checking it across it's own database. The app is almost accurate when you play the song. Might not work as expected when you sing into the app. Shazam is an alternative.
Zedge
Zedge is a one-stop shop for ringtones and wallpapers, all for free. The app is easy to use and has ton of options to personalize your phone.
Queue Tube
This app lets you play Youtube video's audio. Let me rephrase. You can queue youtube videos into this app so that only the audio from the video is played. This facilitates multi-tasking when trying to listen to a song on Youtube.
Ustream
Ustream is for the amateur journalist out there who would like to stream live videos from their phone over 3G. The video quality is mediocre, yet will suffice.
Miscellaneous
Following is the list of apps I found useful and whose name is self-explanatory, hence no description is required.
Battery Status ( On Notification bar)
Auto Rotate Switch
CNN
Engadget
Cnet TV
Google Goggles
The list can go on and on. The more you use your device, the more interesting apps you ought to discover.