I've spent a lot of time going through the android application store (aka Google Play), searching, downloading, reviewing, and usually deleting new apps. Google Play has hundreds of thousands of apps, and is constantly growing in size. The problem is that most of the apps are garbage and not worth downloading. Obviously not everyone is going to like the same apps as the next person and not all phones and versions of Android are compatible with all apps. I'm going too tell you my favorite apps, in no particular order, and you can do whatever you want with the info.
I'll start by saying that I have a rooted Samsung Galaxy S i1900m. It's a Canadian model but it's the same hardware as the American and international models. I have flashed to DarkyRom running Android 3.36. It's not the newest version of Android but it runs most of the apps on Google Play.
Titanium Backup - Allows you to backup all of your applications to your phone or a cloud service. Great if you mess around with your phone a lot like me. Note that this app has extra functions if your phone is rooted.
Opera Mini - the stock Android Internet browser is smooth and fast but lacked the viewing functions that I like. This version of Opera doesn't do flash but makes up for it with pure speed.
Screen Filter - allows you to apply a special filter to your screen to dim the brightness beyond the stock setting. This is a lifesaver if you use your phone in a dark place and want to save your eyes.
Avast Anti-Theft - this app actually installs as a full mobile security suite but the firewall and scanners can be uninstalled to leave you with a great anti-theft app. Note that this app has extra functions if your phone is rooted.
Juice Defender - I purchased the extreme edition, which after making dozens of adjustments, has provided my phone with an average of 86% more battery life. Note that this app has extra functions if your phone is rooted.
GO Launcher EX - the GO suite of programs provides easy to use replacement apps for the stock launcher, messaging and calendar apps. All of the GO apps give extra functions and a ton of themes.
Baby Rattle with Child Lock - your stuck in traffic and your 12 month old is strapped in the rear seat, screaming bloody murder... Out comes Baby Rattle! This is the final answer in keeping your screamer at bay. Note that baby slober can be hazardous to your phone.
I use a lot of other apps but the above aren't as common. Maybe I'll make a list of my favorite Android games at some point. Do you use any great apps that I haven't mentioned?