In order as I view them on my phone. Many of them are available on both Android and iOS. All should be free or have a usable free version:
PicSpeed Wallpapers - A fantastic app for finding high quality, HD wallpapers for your background
Astrid Tasks - A simple task manager, which can be obtrusive if you want or passive if you don't.
John NES - Allows you to play old 8-bit Nintendo games you load onto your device
FBReader - The only e-book read I use. There are many pre-installed locations you can download free e-books from and it also recognizes e-books you manually put on your device
Dolphin Browser w/ Dolphin Jetpack - It's the web browser I use and I'm quite happy with it
ES File Explorer - Almost a must have. It allows you to browse through the files on your device
Angry Birds/Fruit Ninja - Because it just wouldn't be right to have a device without Angry Birds and/or Fruit Ninja
SSG 2 (Super Stick Golf 2) - Very fun game, kind of like a miniature golf style game in terms of having to hit around obstacles
Pulse - An easy way to view news from a variety of sources you choose
Shazam - If you're into music, you probably already know what this does. Basically, it can "hear" a song and tell you what it is
Snapchat - Don't understand the appeal myself, but it apparently is quite addicting. Snapchat allows you to take a picture, edit it and send it to a friend, but after a certain time (I think the max is 10 seconds), the picture is no longer available to be viewed
Speed Test - Great app for testing the speed of your Internet connection
Old Time Radio - This is an app which allows a person to listen to the old radio program broadcasts (things like Gunsmoke, Amos and Andy, Dragnet, etc.)
Netflix - Already mentioned, and probably pretty clear as to what it does
YouTube - Just like Netflix, people probably already know what this does (and usually will be installed by default on Android devices anyways)
Pandora - Streaming music app
TuneIn Radio - Allows you to listen to radio stations from all across the country
WiFi Analyzer - Allows you to view information about Wi-Fi signals at your present location (and will update as you move). This allows you the ability to see if you have any interference with your signal, or where the strongest public Wi-Fi signal may be coming from.
Zedge - An app which kind of serves as a hub for various other things (such as sound tones, backgrounds, other apps, etc.). It's a trustworthy app, which is partly what makes it so appealing considering the function it serves
Paid:
Swype - I'm only going to list one paid app, and it's only $.99. Swype is without a doubt my favorite keyboard on Android and most people I know who use Swype (or the more expensive Swiftkey) agree that after using Swype, manually typing in letters is just not an option anymore.