I've had to give out recommendations for apps for Android tablets three times now since Christmas, so I thought I'd just write up what I consider some "must-have" apps.
Note: I've left off some of the more obvious ones (Gmail, Dropbox), games, the customization ones, and the ones that require rooting or any kind of tech skill. That said, feel free to suggest your favorites (with a link) below.
https://play.google.com/store/apps/details?id=com.flyersoft.moonreaderp Moon Reader closest thing to a decent ePub reader. (The official Kindle app is the only decent mobi reader I've found.) https://play.google.com/store/apps/details?id=com.trello Trello: Corkboard-style organization. While it has a pro version, the free version is awesome enough, and you can access it from any browser as well. https://play.google.com/store/apps/details?id=com.google.android.apps.googlevoice Google Voice: You will be able to text over wifi. You can get a new number or port your existing one. https://play.google.com/store/apps/details?id=com.ideashower.readitlater.pro Pocket: Save and read stuff from over the web in an simple easy to read format https://play.google.com/store/apps/details?id=net.alliknow.podcatcher Simple Podcatcher: Lets you stream podcasts. You don't store them locally, so you don't have to worry about the size of your SD card. https://play.google.com/store/apps/details?id=com.twentyfivesquares.press&hl=en
Press: Highly recommended RSS reader. Works with Feedly on the backend. https://play.google.com/store/apps/details?id=com.sand.airdroid AirDroid: Lets you transfer files easily to your tablet from your PC over your LAN. https://play.google.com/store/apps/details?id=com.andmadesoft.share Andmade Share: Configurable "share" menu so you don't have to see all the stuff you don't WANT to see. https://play.google.com/store/apps/details?id=org.connectbot Connectbot: Pretty decent SSH client. If there was only a way to have tab completion work.... https://play.google.com/store/apps/details?id=com.contapps.android Contacts+: One of the best contact apps I've ever seen. EVER. Pulls in from where you want it to, and doesn't screw up existing contact settings. Very nice. https://play.google.com/store/apps/details?id=org.deadbeef.android Deadbeef: Just like the *nix Deadbeef player, it handles about any audio format (including streams) you throw at it. https://play.google.com/store/apps/details?id=com.mvilla.draft Draft: If you actually want to write (anything other than code) on your tablet, this is the app you want. Syncs with Dropbox, supports markdown. https://play.google.com/store/apps/details?id=com.aor.droidedit.pro Droidedit: Excellent for writing code.
https://play.google.com/store/apps/details?id=com.android.keepass Android Keepass: Compatible with KeePass and KeePassX with dropbox syncronization, so you can have your passwords secure AND portable. https://play.google.com/store/apps/details?id=com.levelup.touiteur Plume: I like Plume for my power-user Twitter needs
https://play.google.com/store/apps/details?id=com.vpn_menu_pro
https://play.google.com/store/apps/details?id=de.blinkt.openvpn VPNs are pretty much required these days... I can't just have OpenVPN due to some firewalls, hence two VPN apps. https://play.google.com/store/apps/details?id=the.pdfviewer3 PDF Viewer: Views PDFs as it says on the tin. Does it well. https://play.google.com/store/apps/details?id=com.mint I didn't "get" Mint when it wasn't right there and easily accessible. Mint makes sense on a tablet, especially with a widget.