I'd like to know if you know of any better (preferably free) apps than what I am currently using; and, if you do, tell why the are better?
Also, this may help those new to Android to find some basic apps to start out with.
I HIGHLY recommend rooting your phone to gain access to Ad Blocking and other minor conveniences and customizations.
NOTE: These are on my Droid1. I am running Stock, Odexed Froyo 2.2 FRG22D (MetickOne's). DroidX/2 may have better stock apps so this list probably won't apply fully.
(* requires ROOT access)
ADWLauncher / Launcher Pro: Home (Launcher2) replacement. Add extra screens and dockbar buttons and lots more. Both are way better than the limited Stock (Launcher2). I personally use ADW because it has been more stable and has more themes; although, Launcher Pro has more features. Launcher Pro also has a paid version that adds some useful widgets.
* AdFree: Updates your .hosts file to block common ad sites. I like using this because they provide updates every month or so.
Agenda Widget: Syncs to Google Calendar automatically and provides a highly customizable, scrollable widget. I use this to display a 2x4 calendar agenda on my homescreen.
AppRemover: One of the fastest and easiest apps to let you quickly uninstall other apps. There are at least 5 others of these that work 99.9% the same way, but this has been my favorite so far.
aTrackDog: Manually check for the latest versions of all your apps (including non-market ones) and one-button click to update them.
Bubble: simple bubble level.
* BusyBox installer: Manually and conveniently checks for the latest version of BusyBox (only needed for ROOT users)
Camcorder: Creates a 'button' that takes you directly to the Camcorder. Simple, nice shortcut.
ConvertPad: Simple measurement converter.
Currency Converter: Simple currency converter.
Date Status Bar: Adds permanent (that is, the [CLEAR] button won't remove them) notifications that let you add text to the notification bar (in this case: the date). Simple app that adds functionality that should have been included in Stock as far as I'm concerned.
* droCap2: Simple way to take screenshots of your phone screens.
* Droid Wall: One of (if not THE) only firewall apps. Has black/white list options.
eBuddy: Better than Meebo IMO and best multiple-account IM client to use until Trillian on Android is completed. NOTE: There was a fake version of this on the Market a while back that appeared first in searches.
GTasks / Google Tasks Organizer - Trial: Both connect to your Google Tasks and auto-sync. Great for making quick shopping lists, to-do lists, etc.
Handcent SMS: Vastly superior to stock SMS app. Works and looks nearly the same, just with a LOT more features and highly customizable.
JustPictures!: Connects and auto-syncs to various popular photo sites (including Flickr and Picasa). Simple interface and features that I like much better than stock "3D Gallery".
Last.fm: Streaming music, especially great for listening to specific genres.
Lookout Mobile Security: Android "antivirus" but mostly useful for backing up your SMS/text/data and locating a lost/stolen phone. Whatever the "antivirus" does adds some piece-of-mind I suppose - there are countless other apps for backing up data and finding phones but this one has been the most reliable and easy to use.
MotoTorch LED: Use your screen and LED Flash as a flashlight. Fastest on/off response time and includes a widget.
Music (or MP3) Downloader Pro: Easily search and download MP3 files. Seems legit so far, but I have been a bit paranoid about using it as it finds MP3s so easily. NOTE: Once they are downloaded, moving/copying them somewhere else makes them unplayable on the phone in the new location (somehow?).
mVideoPlayer: Same as JustPictures! but for videos. Doesn't play as large of a range of videos; but, works very well for Android-optimized (non-.AVI) videos. Stock 3D Gallery falls far behind.
* OSMonitor: Provides detailed information about running tasks, network activity, etc.
Pandora Radio: Streaming music app that plays artists/songs that sound similar to what you search (i.e. Metallica Radio will eventually play Megadeth, Slayer, etc as well).
PicSay: Simple way to edit photos directly on your phone. Easier and more reliable than the Photoshop app IMO. I have only used this a few times, though, as editing photos on a tiny screen is tedious at best.
* Quick Boot: Provides a "button" to give you more shutdown options: reboot, reboot into recovery, etc.
RealCalc: Scientific calculator with way more functionality than stock calculator app.
* Remote Wipe: Remotely wipe your entire phone and SD card if it is stolen. It works by responding to a specific SMS message you send your phone. I haven't yet tested this but the comments/ratings say it works. (I think this needed root to nuke your SD Card. Let me know if I'm wrong).
* ROM Manager: Easily install various ROM, Kernels, Themes to your rooted phone. Also has quick access to recovery mode and can update/change your recovery "program" (SPRecovery, etc). I highly recommend the PAID version for extra features and to support the ROM community.
* Root Explorer: File manager for Root users. Alternatively, "Astro File Manager" is great for non-root users.
* SetCPU: Under/Over-clocking app with temperature monitoring, profiles, and a widget. If you want to OC your Droid1, get this.
ShopSavvy: Barcode scanner that can also search for prices and similar, local items.
SpeedView: Speedometer (like in your car) with many other related features. This works and is +/- 1MPH accurate when I use it in my car so it'd be great to have if my car speedometer ever broke.
StopWatch & Timer: Simple StopWatch...and Timer!
* Superuser: Keeps you updated to the latest Superuser version. Only needed if you have Root.
Terminal Emulator: Simulates a LINUX terminal. If you have Root and know LINUX commands then this will be useful - otherwise...probably not so much.
* Titanium Backup: Back up your apps to SD card and much much more. I don't actually use this any more; but, if you are frequently changing ROMS then this will save you a lot of headache. The paid version gives you more convenience options such as clearing DALVIK cache w/o going into recovery mode, etc.
Tip Calc: Simple tip calculator.
Vignette: The most reliable, yet easy-to-use camera app I've used. Many more options than stock camera (timer mode, scene mode, effects, etc). You can also disable the camera shutter sound with this.
Google Voice: Free minutes and texts. Free voicemail. Free visual voicemail. Free extra phone number. If you have an unlimited data plan but limited minutes/texts then you should use this. NOTE: Currently, you CANNOT send SMS picture messages (text only).
WeatherBug: My favorite, free way to keep the current local temperature in the notification bar. Also shows live weather, etc in the app, itself.
Winamp (currently in Beta) / Mort Player: Much better and more customizable than the stock music player. Has widgets.
The great thing about Android is that most of these apps are 100% free (albeit, with ads).
The crappy thing about Android (the Droid1 at least) is that most of these apps are vastly superior to their stock versions (the photo, video, and music player and camera apps especially) so you have to do a bit of work to 'fix' your phone with better 'stuff'. Also, a lot of the basic stuff (converters, scientific calculators, etc) doesn't come with the phone.
Hopefully, this helped you with that 'stuff'.
Please let me know your opinions and/or alternatives or if I left something vital out!