I have had an Android phone for almost 2 years now and I must say, I prefer Android over iOS (the iPhone OS). I have an iPod Touch for all that stuff, and while it's nice and shiny and has a far better range of apps available for it, Android is still my favourite mobile device OS.
And so, here is a list of some of my favourite apps:
My phone (HTC Wildfire) has a smaller screen than a lot of Android devices, and as such a lot of the mainstream Android apps either don't work or simply aren't available for it. The official Twitter app for Android is one of these - it works, but the smaller screen means half the options are cut off and it makes it very difficult to use. For this reason, I chose Twidroyd - I used to use it before Android got an official Twitter app and I really liked it. Twidroyd has all the functionality of the official app and more - you can integrate Identica, use multiple accounts, use bit.ly API keys, upload to a variety of picture and video hosts, 'mute' especially noisy tweeters and more. Some of those features (namely multiple accounts) requires the paid-for version, but the free (ad-supported) version has more than enough features for the obsessive tweeter :)
Edit: Twidroyd is no longer available after being replaced by Ubersocial. I wouldn't really recommend Ubersocial as it's rather too full of ads for my liking. The official Twitter client is much better :)
Android System Info
This is more of a geeky one - if you want to find out a load of info about your phone, like what sort of features the processor has or what versions of OpenGL it supports, this app will tell you. I mainly use it to keep track of how much space I'm using - the bar charts on the front page will tell you all that and more. It also comes with a task manager, app manager (like Android's built in one, but this gives you more at-a-glance info - like how much space is being used by caches - than the Android one) and access to the system logs.
Google Maps (Navigation)
Ok, I know this comes on most (if not all) Android phones, but they don't always have the most recent version - the one that has the free turn-by-turn navigation app bundled in it. Before this, I paid a monthly fee for a really rather nice GPS app - definitely rivalled the dedicated units in shops. Google's Navigation app isn't quite as nice as that, but it does the job very well for a free application and I'm much happier using that than buying a full GPS unit or paying a monthly fee.
The Dropbox app for Android isn't as good as the one for iOS, and is definitely nowhere near as good as the PC/Mac clients, but I find it very useful to share photos from my phone. My last phone had an accessible SD card slot, where I could take the card out and read all the pics on it - this one doesn't, so if I want to grab the photos I need to pretty much take the phone apart. Not very convenient. This way, I can upload my pics straight to dropbox and they sync to my computer automatically. Lovely!
Not sure if this one is downloadable, or if there is a similar downloadable app - this one came with the 2.2 (Froyo) update to my phone, courtesy of HTC. It allows me to broadcast my phone's 3G connection over wifi so that I can use my laptop even if there's no connection handy. Great idea for when I'm out and about and don't have to pay extortionate rates for hotel wifi :D It can be locked down with WPA and WPA2 security as well as only allowing a certain number of devices to connect.
Other things I haven't included here but use every day are: Facebook, Internet, Weather and Gmail, but those are all built-in apps (apart from Facebook, which is nothing special in terms of the app) so I'm sure most people already know about them :P
Anyone got any other favourite Android apps? Always open to try new things - and if they don't work on my Wildfire, I have a rooted G1 that things will definitely work on :D