A few of my favorites.
Right now I'm very impressed with AirDroid.
https://market.android.com/details?...1bGwsMSwxLDUwMCwiY29tLnNhbmQuYWlyZHJvaWQiXQ..
Network Signal Info works great if you want to check your network connection strength.
https://market.android.com/details?...=W251bGwsMSwxLDEsImRlLmFuZHJvaWQudGVsbmV0Il0.
Brightness Profiles is my favorite for setting up and easily controlling my screen brightness settings.
https://market.android.com/details?...yeWRvdWdobnV0cy5hbmRyb2lkLmJyaWdodHByb2YiXQ..
Android Lost for lost or stolen phone.
https://market.android.com/details?...t#?t=W251bGwsMSwxLDEsImNvbS5hbmRyb2lkbG9zdCJd
Parkometer for keeping track of the time on parking meters and finding your car if you don't remember where you left it. This has become more of a problem as I've gotten older.
https://market.android.com/details?...sImRlLmphbnNjaG9wcGVuaG9yc3QucGFya29tZXRlciJd
PlayerPro is my favorite music player at the moment. I try others and always go back to it.
https://market.android.com/details?...=W251bGwsMSwxLDEsImNvbS50YmlnLnBsYXllcnBybyJd
I like mahjong and Random Mahjong is my favorite.
https://market.android.com/details?...zaG9lY2FrZWRyb2lkLlJhbmRvbU1haGpvbmdQcm8iXQ..
Backup (I'm not rooted), My Backup Pro
https://market.android.com/details?...vbS5yZXJ3YXJlLmFuZHJvaWQuTXlCYWNrdXBQcm8iXQ..
Password Manager, LastPass.
https://market.android.com/details?...sMSwxLDEsImNvbS5sYXN0cGFzcy5scGFuZHJvaWQiXQ..
Bookmarks, XMarks, now owned by LastPass. Lets me sync and access all of the bookmarks on my computer from my Android phone.
https://market.android.com/details?...1bGwsMSwxLDEwMiwiY29tLnhtYXJrcy5hbmRyb2lkIl0.
Best weather app I've found is Weather, Radar, Alerts, Quakes by Elecont software.
https://market.android.com/details?...sMSwxLDEsImNvbS5FbGVjb250LldlYXRoZXJDbG9jayJd
Volume Control + Pro if you want to control and lock your different volume settings.
https://market.android.com/details?...1bGwsMSwxLDEsImNvbS5jYi52b2x1bWVQbHVzUHJvIl0.
Favorite grocery list is Mighty Grocery. I love being able to just speak what I want and have it added.
https://market.android.com/details?...sImNvbS5taWdodHlwb2NrZXQuZ3JvY2VyeS5mdWxsIl0.
Keyboard. I didn't use to care for swype type keyboards, but it didn't take long for Flext9 to become my favorite keyboard and I've tried most of them.
https://market.android.com/details?...sMSwxLDEsImNvbS5udWFuY2UuZmxleHQ5LmlucHV0Il0.
Browser. Lots of good ones, but I still like Dolphin HD the best.
https://market.android.com/details?...sMSwxLDEsIm1vYmkubWdlZWsuVHVubnlCcm93c2VyIl0.
Calendar: Business Calendar
https://market.android.com/details?...=W251bGwsMSwxLDEsIm1pa2Fkby5iaXpjYWxwcm8iXQ..
Scheduler/Planner: Pocket Informant. I like the different views it gives me. It's always synced with Google just fine for me. I hope they make it compatible with ICS before I get the update. Kinda' pricey, but it's about half price right now on Amazon. ($4.99)
Amazon.com: Pocket Informant: Appstore for Android
I almost forgot Taskos. Still playing with it since it's fairly new to me, but I do like it so far. Another one that uses speech to text.
https://market.android.com/details?...esult#?t=W251bGwsMSwxLDEsImNvbS50YXNrb3MiXQ..
Network connection app is LTE onOFF. I know Verizon says not to use it, but it gives you more options than their built in network switch and is basically a shortcut to dialing *#*#4636#*#* on your keypad which takes you to Verizon's own advanced settings.
https://market.android.com/details?...1bGwsMSwxLDEsIm5ldC5hbmRpcmMubHRlb25mb2ZmIl0.
LastPass and XMarks are bundled for $20.00/year. To me it's well worth it, but others may disagree.
Probably one or two more I'm forgetting, but I use these every day. Most of these are the Pro, or paid versions, but most have free versions available. If I like an app I try to buy it and support the dev.
Hope these links work.