[AT&T Samsung Galaxy Note I717] carricklongley:
I thought I would add in my two cents here for those that are new to Android. My one word of advice is don't be afraid to pay for apps - there are some amazing applications that are the cost of a soft drink or beer and they really make the experience much better.

Utilities and System

Adfree (requires root) - blocks Ads across most of the phone

aDownloader - Torrent downloader

Airdroid - great app to remote in to phone, transfer files, etc.

androidlost - track missing phone (not convinced this is working properly on Note)

androidVNC - great VNC app

Barcode scanner - scan QR codes and standard 2D barcodes

Beautiful widgets - it is what is

Camscanner - take photos of documents and convert to PDF (awesome)

Connectbot - SSH client for Android

Dropbox - awesome

Dropsync - makes Dropbox actually a 2 way sync - worth the $$

ES File Explorer - great file system browser / utility

LCD Density Modder Pro (requires root) - lets you easily change LCD density

LogMeIn - remote control computers from your device - free service

Root Explorer - a staple for any rooted device

Screencast (requires root) - make screencasts from your phone

Speedtest - it is what it is

Titanium Backup - the Swiss army knife for rooted devices

Ubuntu installer (root) - run a full version of Ubuntu on your phone (via VNC)

Ultra GPS logger - awesome GPS utility

WiFi tracker - great utility for spectrum analysis of WiFi hotspots

Games

Dead Space

Golf Extreme - Easy golf game, but great for killing time

HomerunBattle3D - OK baseball game. Similar to Golf Extreme (currently cheap!)

Grand Theft Auto 3 - A ported version of GTAIII, runs great

Shadowgun - great shooter game

(As you can see, I am not much of a gamer)

Media

doubleTwist - for those having iTunes withdrawal or if you like a clean interface, doubleTwist with Airsync are great

Flixster - Check movie times, reviews, trailers

IMDb - the famous international movie database on your phone

MX Video Player - an awesome video player that can handle most every codec

Pocket casts - fantastic podcaster with scheduling and other options

Soundhound - great way to listen and find out the name of 'that song'

TuneIn Radio Pro - stream radio stations over IP on your phone

Others not categorized

Here are some other great apps I have that I didn't stick somewhere above. They are either fairly well known, or you will likely have come across them on the Market, I mean the "Play store."

Amazon - if you are not sure what this is, welcome to Earth.

Amazon Kindle - its the Kindle app

Antennas - log RF signals from both cellular and WiFi hotspots near you

Camera ZOOM FX - Decent camera replacement app

Craigslist - Decent Craigslist app

Currents - Pretty neat magazine like app from Google

DavDrive - Access your phone via WebDAV; similar to Airdroid, but I find it simpler to use.

Google Earth

Evernote - Good note syncing app

Facebook - Yep

Googles - Take photos of objects and Google can search their database for what it is

Google+

GPS Status - Another GPS utility

GPS Test - Another GPS utility

ICS Keyboard - Great keyboard replacement for the stock one in my opinion

imo - Great multi-chat client (free too!)

Marriott - If you stay here, actually not a bad app

Netflix - I don't pay for their service anymore, but if you do, you likely have this

NPR News - Great app for listening to NPR and reading some of their featured stories

OpenSignal - Another signal mapping utility

OpenTable - Reserve a table at a nearby restaurant right from your phone

Pandora - Awesome music discovery tool

PayPal - Yep

Quadrant Standard - A great way to measure how fast (or slow) your phone is

RF Signal Tracker - I like RF signal tools, but I am an engineer so there you go

RunKeeper - Great app for running and tracking your exercise

Skitch - OK app that works with Evernote to take advantage of the s pen

Skype - Hopefully Microsoft doesn't ruin it

SwiftkeyX - Predictive test keyboard replacement for Note

SwitchPro Widget - Create widgets that let you do lots of things

Unit Converter - 'Nuff said

WebSharing - an Airdroid like app, I don't use it very often, but it is an alternative