One comment: I don't really use task killers as Android does a very good job of managing tasks and memory. But - occasionally - I do resort to killing an app I specificially do not want running at all. In that case, I use Advanced Task Manager (paid).
My fave apps:
AndFTP and SwiFTP - the first is an FTP client and the second is an FTP server. A great pair for easily moving files between devices of *any* kind as long they have either an FTP client or server. Both free.
Estrongs File Explorer - a great file manager, but can also access Windows / samba file shares in a non-domain setting. Free.
AndroZip - A general archive management (rar, zip, whatever) utility that doubles as a very useful file manager. Free.
Rock Player - the only media player I know that can play Xvid AVI files on Android. Free or paid.
Meridian Pioneer - my favourite all-round music / media player. Free or paid.
Antropia - An media player and streaming Internet radio app. Average UI but *key* feature is that it buffers what you're hearing to the sdcard and you can save it as an mp3 if you liked it. Free (I think...not sure).
Beautiful Widgets - A Sense-like skin-able (loads!) clock and weather widget plus many other functional widgets (wifi, GPS, Bt, etc...etc..) Paid app
Handcent SMS - best SMS app. Period. Free.
"Maps and Nav [brut]" - turn-by-turn GPS nav. Hacked version of Google Maps. Effectively separate app. Exists alongside and does not conflict. Works really well. Free. Can be obtained via XDA-Developers and various web sites. A must have if you want GPS Nav for free.
AndCam3D - take 3D photos (stereo or anaglyph)
SlideIt keyboard - paid app. It's a slide-ya-finger-to-type keyboard. Easily as good as "Swype"...and not beta. It's a finished product and you can buy it. Better than ShapeWriter. I love this app. It makes mot typing easy and one-thumb, one-handed typing becomes possible and practical.
Chrome to Phone - get the Chrome browser extension and the Android app (free)...and send interesting web links to your phone from your Chrome browser. Click "send to phone" on the PC in Chrome...and then open the browser on the phone and the page you sent will come up on your phone.
SMS Mail Monkey - "Thanks for your txt. I'm not in right will. Will text you back!" - or whatever. Answer phone for txts.
Backup to Gmail - backs up txts, MMS and your phone's call log to your gmail account, filing each under an appropriate label. Love this app.
Better Terminal Pro - the best linux shell / terminal app for Android.
Wifi Tracker - paid app. Walk or drive along with the app running and GPS enabled...and it detects, locates and logs all wifi it sees. You can export CSV, KML (Google earth) and another format....to the sdcard or to email. The Google earth files are awesome. You load the file and all the wifi is geo-located with clickable tags that dispay all attributes.
Camera ZOOM FX - loads of effects and props. A major extension of function beyond the default camera. Paid app.
Picsay Pro - take a photo, "share" it to PicSay Pro -> add a caption, effects, titles, adjust images properties (brightness, conrast, etc..)...and then save to SD or "share" it onward to any app thqat can receive a share -> gmail, twitter apps, printershare....*whatever*. Paid app, but I love it.
Printer Share - print stuff from your phone to any printer connected to a PC with the Printer Share PC app on it. Eg. - Can print on printer at home from phone anywhere. First 100 pages free.
Clipbot - supplements phone's clipboard. When the collector is on it captures and keeps all cut or copied text for individual or group selection later from the Clipbot app. Very useful when trying to copy / paste several pieces at once in stead of treating each individually.
SPB TV - Paid Internet streaming TV app. Must be over 100 channels from all parts of the world. Beware the Polish shopping channel that plays porn ads all night - in Europe. Includes Bloomberg, Euronews and many more.
TransDroid - Easy, simple app for remotely managing a bittorrent client / server.
AppMonster Pro - I use this for backing up the APKs for non-protected installed apps.
Ping - free. It pings things.
JetFlicks - subscribe to on-demand streaming of many of the most popular TV series of recent years. Usually multiple seasons available.
Adobe Flash Player - free. Needs Android v2.2. Works great with TVNZ OnDemand. Doesn't work with TV3's on demand. (Can see the ads....but the show never comes up).
Last but not least: Skyfire browser. Allows you to watch flash video on phones that can't yet support the Flash. I use d it mainly to watch video clips on the BBC News web site.