Astro File Manager - The app helps you to organize files in your Galaxy S III. It's like windows explorer for your phone and allows to browse all the files on your phone memory and sd card. You can also view pictures, music, videos and documents while organizing with this app. The features of the app includes file management, android file browser, app backup, image and text viewers, SFTP, Zip/ Tar, downloader, search files, application manager, task manager etc.
WhatsApp Messenger - The app is a smartphone messenger for your Galaxy S III. With this app you can connect to your friends who are using not only Android phones but also iPhone, BlackBerry and windows phones. With this app you can send videos, images, voice notes and also can group chat with your friends. This is a must have app for Galaxy S III to stay closer with your friends.
MXPlayer - This is No. 1 video player available on Google Play and I personally prefer this player since it plays almost all the videos which didn't play in vlc, mobplayer on my Galaxy S III.The player also supports multi-core decoding and provides highly optimized codecs for CPU. It also has pinch to zoom, subtitle scroll, kids lock support.
SwiftKey 3 - SwiftKey 3 app is one of most used app on Galaxy S III. It understands how words work together, giving much more accurate corrections and predictions than other keyboards. You can even personalize the app by giving your facebook, twitter accounts. The free app will work for 1 month with all the features. Try the app, I am sure you will like it.
App 2 SD - App2SD is a simple that does what it was mentioned on the title. With this app you can view all SD card capable apps and move them with ease. This app saves space further by clearing all the cache of applications. You can also quickly uninstall any app by drag n drop.
Opera Mobile - Opera Mini for Galaxy S III is the fastest browser which compresses data by up to 90%. It is the best choice when using slower networks or paying per megabyte of data used. The app is smooth and speedy. There were many features like pinch-to-zoom, bookmarks synchronization, speed dial etc.
Fruit Slice - Fruit Slice is the best free fruit juicy game on Google Play. The game has many modes to play and it will entertain you for hours. You need to just slice the fruit by your finger to play.
Antivirus Free - Antivirus Free, powered by AVG detects harmful apps and text messages. It is a mobile security solution that protects your mobile device from spyware. The app has many features like scanning apps and media in real time, locate your lost phone, lock and wipe to protect privacy, kill tasks etc.
SoundHound - This app for Galaxy S III recognizes the music. It features world's fastest and most accurate music identification. You can also get live lyrics for your favourite songs. It also has real-time facebook and twitter updates from the artists of whose songs are being played.
Tweedle - Tweedle is a free twitter client for your Galaxy S III. It has clean and fast interface. You can also select color scheme. The app is relatively new in the market and will be soon updated with tons of new features.
To download any of the above app, you can also just search for the app in Google Play on your mobile.
Also check out the 30 Android apps that I was using earlier on HTC Desire HD.