Here are some apps that I think are awesome for any Android user - not just rooted users or SGS2 users.
Beautiful Widgets ($2.79) - I was looking for a nice-looking weather widget and - literally - every single weather widget in the Android Market is terrible except for this one. Simple and clean; I've got a 1x1 tiled widget under the calendar on the left-hand side (click for big):
Also pictured: Widgetsoid, taking up the rest of the space under the calendar.
Also, plain black background is the way to go!
Cubed (^3) Music Player (free) - A very smooth and intuitive music player. Highly rated and well-reviewed on the Android Market. Much better than the stock music app on Samsung Galaxy phones.
That's pretty metal.
DelayedLock ($1.49, or try the trial version) - Tired of re-entering your passcode every time your display turns off? DelayedLock does exactly what you think - adds a delay between when your screen turns off, and when a passcode is required. Mine is set at 10 minutes.
NoLED (free) - this one is for any Samsung Galaxy S phone. So awesome I made a whole blog post about it. Every person I've told this app about loves it, so just trust me!
No Wallpaper (free) - Sets the background to completely black. Blacker than the blackest black, times infinity. On AMOLED screens, like the Galaxy phones have, black pixels don't draw a current, they're just "off". So if you're flicking around your home screen a lot, this will save you a bit of battery life.
Smooth Calendar (free) - The best calendar widget I can find - one row tall by four columns wide. Pictured above on my home screen. Customizable and it gets the job done! It could be sleeker though.
WhatsApp (free for 1 year, $2/year afterwards) - excellent messaging app. Works over WiFi, so it's ideal for my crappy basement apartment where I get no signal. It works a lot like BBM, and it's available for Android, BlackBerry, iPhone, and Nokia phones!
Widgetsoid (free) - Great app for making custom homescreen widgets. See the first image in this post - I have a 1-row, 3-column widget with shortcuts for airplane mode, screen orientation lock, flashlight (using camera flash), system volume controls, and a battery monitor!