These are some of the apps that I've come across and use that are absolutely essential to the well being of my android phone. I currently use a Mytouch 4G rooted with Cyanogen 7.1 Release Candidate so some of these apps may require root. If I don't mention it's cost, the app is free.
Info: An app that I install first and foremost whenever I'm working on someone's phone. Has an array of tools to keep your android running in top form.
Info: While this one isn't truly essential, it's useful if you need to use Java on your phone. It allows you to connect and use a virtualPC running on a remote server. The PC runs Fedora and has Java and Flash with Google Chrome, Open Office, and Adobe Flash installed. This is for both iPhone AND Android and costs $20.00. A deal and a half in my opinion.
Dolphin Browser HD:
Info: The greatest Broswer on Android. Supports Flash and doesn't afraid of anything. Looks great on an HD screen and loads pretty quickly.
Info: A useful app for browsing your phone's memory and sdcard. Can extract zip and rar files and delete apps forced on you by your carrier. There are free alternatives but this one is the best in my opinion.
Norton Mobile Security BETA:
Info: A new and currently FREE malware protection app for Android. These days, Android malware are emerging more and more so it's about time to take up arms and install some sort of anti-malware application.
Info: An app that installs truly beautiful widgets on your phone. I use it only for the "Super Clock", which makes your phone look pro. There is a free version.
Info: An alternative launcher for Android. Speedy, customizable, stable (most of the time), and cool looking. Definitely essential. I have Launcher Pro PLUS which is a paid version but is even more customizable. I'll post some screen shots at the end of this post.
Info: THE greatest music app for Android. Supports a lot of different music formats such as FLAC and ALAC. It's widget looks awesome, the app it's self looks awesome and it even has an Equalizer! I can't listen to music on any other app but this and Pandora. It has a free trial that is just a free version that bugs you forever that you haven't paid yet. In some instances it may stop working, but it hasn't done so to me yet.
Info: A CPU governor app that allows you to overclock your phone if your kernel supports it. Useful for keeping your phone running fast when your using it while keeping the battery usage to a minimum when the screen is off. It costs money but is totally worth it.
Noom Weight Loss:
Info: An app that keeps track of your weight, a daily log of your Calorie intake, and a fitness program that can keep track of your walking/running and accurately predict calorie burn of all sorts of excercises. Even if your not trying to lose weight, a perfect app for keeping up your health! I've personally lost 22 pounds using this app and I recommend it =D
Info: An App for boosting the speed of your SD card. It works by increasing the cache of your SD card. Absolutely free and useful.
And Here are the screenshots I promised! I'm using Launcher Pro Plus. Widgets on the screen include Beautiful Widgets Super Clock, Youtube Widget, and the SetCPU widget.
And here's my app drawer. Launcher Pro has the 3D App Drawer option, as you can see in this screen.