EDIT 2017: this is a post from my old blog, kept it for the nostalgia.
These are the ones I can't live without. Obviously I wont include the Facebook app because everyone knows about that, these are just the ones that I personally use every day and think are worth a download.
Brilliant Twitter application, handles all your 'tweets' and is very fast. My favorite part is the search, where you can set the application to find people around your location; a little bit stalker-ish but fun.
2. FC Tasks-
A mobile app by Franklin Covey, maker of stylish day-planners, which sorts all your tasks by date. I have used a lot of task manager apps and this is certainly the best one purely because of the amount of features it has. You can organise your tasks by date, and category, and then by order of importance. It is clear a lot of work has been put into this app and it looks brilliant, but there is a price to pay for such a great application, and that is £3.49.
Sign up for a Last FM account online and you can use this brilliant app. You basically search for an artist and it generates a 'radio station' for that artist, streaming songs which are a similar style to that artist (requires wifi). And it works really well. Great for when you are uncertain of what to listen to, and most of the time the app will introduce you to new artists. It's also free, which is hard to believe considering how well it works and the features available. Highly recommended.
4. National Rail-
A boring one I know, but the National Rail app is brilliant for when you are travelling by train. It allows you to add stations of your choice and create a route planner with live updates. It also has a location feature in the search, and home stations. It may seem steep at £4.99, but if you use rail travel a lot in Britain (and know how terrible it can be) live feeds are very useful.
Many apps offer news, but I like the way The Telegraph app offers many categories and is quite fast. I used Wall street journal, ITN, BBC reader and a couple of others but I found this to be the best.