Allow me to save you a lot of time since I am the designated app investigator for my entire family, and have tried a ridiculous # of apps out there:
Browser:
ICab Mobile is the best I have found (runners up: Atomic, Perfect Browser)
Essential Utilties:
1password - good luck cracking my 25 character indecipherable passwords - a really good idea to use 1password, and the ipad app is top notch
Evernote - incredible, free, note taking apps for your computer and ipad
Dropbox - syncs many apps and your computers with ipad, this like Evernote is an absolutely essential service to run full time on all your machines
System - just your basic system info app but very useful - memory, batt, etc
Free Wi-Fi Finder a free hotspot finder that actually works really well and allows you to download locations for viewing w/o internet. Most of us have either 3g or Mifi, but it can still help in a jam
Air Display - allows you to use your ipad to extend your laptop screen, only necessary if you need such a functionality, but very cool. An example would be if you use Photoshop you can put your brushes on your ipad screen and have them available at your side so you can maximize screen space for whatever you are working on.
Video:
Netflix - if you don't have netflix streaming, then get it immediately
Air Video - one of the best apps ever hands down - streams tons of video off your local machines/servers to your ipad)
VLC- I usually use the native ipad video app but this can handle most other video - not as good as the computer version but still good and likely to get better)
Photo/Drawing:
ArtStudio - drawing, editing
Photogene - photo editing
Adobe Ideas various
Diptic - photo layouts
iChalk - rarely bother with these gimmicky type apps but this one is special
Astronomy:
Nasa HD - they did a real nice job with this and probably just will get better
Observatory - lots of scientific data
Star Walk - this is of course one of the most famous apps and very cool
Planets HD - kinda of the Star Walk for planets
Read:
Flipboard - !!! doesn't really get better than this free app imo, you create categories and it aggregates a web-based magazine for each topic. The elegance of this app is amazing, there is a reason it was app of 2010
Skygrid - similar principle to flipboard and I happen to like this one too
Kindle - duh
Discover - applies flipboard like principles to wikipedia entries, actually very awesome and I think free? app
SI Football - if you are a football fan, this is a great example of interactive books on Ipad
News:
BBC, USA Today, CNN - any of these are decent, I usually prefer the webpages directly but that's just me - i always go to ICab Mobile first
NPR - this is a pretty special app and has a lot of interactivity. Also it's free of course
Music:
Pandora - just about the greatest app ever no matter what device you are on
SoundHound - I personally use soundhound all the the time and have always found it better than Shazam and worth the money)
Aweditorium - really good interactive music magazine thingie
Albums - I wouldn't say this is a necessary app by any means but it is quite cool and kinda a show off app if you are proud of your music collection
Movies:
IMDB - again I usually use the website but this is a good app for what it is
Flixster - great great way to look up movies and links to Rotten Tomatoes reviews as well
What's On? - Essential, great app that shows TV listings for basically any service anywhere... I never realized how much better it is to have a guide at your finger tips instead of endlessly scrolling through the on TV menus.
Shopping:
AppShopper - great alternative to app store surfing, has a lot of extra functionality
Target, Ebay, Zappos - these are all examples of companies that made decent apps vs a lot of retailers that couldn't be bothered
Windowshop - truly great shopping app by Amazon which is kind of a shopping free for all
Deliveries - tracks UPS, Fedex, etc with push notifications, etc.. amazing
Travel:
FlightTrack Pro - best overall flight app, kinda pricey tho, this one depends on how often you travel.
Lifelike Clock - my favorite alarm clock
Weather HD - a popular weather display
Kayak, Tripadvisor, booking.com - again good options for travel website apps
Wikihood - cool geo location based info app
FlightBoard - pretty damn cool, displays arrival departure info for any airport in the world essentially, so no battling other people to see your info when traveling
Other:
Epicurious - my fiancee uses this non stop for recipes and it is generally regarded as the best food app
5-0 Radio HD - I think this app is awesome, basically police bands from anywhere and a lot of bonus features
Magic Window - One of my favorite apps, just download it for 99 cents and you won't regret it. It basically displays relaxing scenes
Fire HD - another awesome one that simply turns your ipad into a crackling fireplace
Remotes: (use these to control my Home Theatre and network computers)
Rowmote Pro - simple, long established remote app that seems to always just work better than the rest although I own A LOT of remote apps
Remote HD - this would be my runner up to rowmote pro, of course these apps are mainly used for controlling your computers, especially if you have them connected to your home theatre
Sonos for Ipad - if you have sonos then this was probably your first download
Splashtop - probably the simplest way to view your computer desktop when you are on the same wireless network
TeamViewer - install TeamViewer on all your computers and your ipad right now. Take a few minutes to set it up and you will be able to control your computers from anywhere in the world.
Wake - very specialized app for waking your computers on Lan signals out of sleep mode. If you don't know what I am talking about then move along, if you do then download it.
Games:
Words with Friends HD - like Scrabble but so much better
Monopoly HD - this game is amazing, play games of monopoly in 30 minutes instead of 3 hours does suck that it's $9.99 with no current discount
Space Invaders Infinity - extremely impressive arcade game
Angry Birds, AB Seasons, etc - everyone knows to get these although they are far from my favs
Solitaire HD - just beautiful solitaire, blows the rest away but costs $$
Risk, Yahtzee, Snood, Boggle, etc - buy all the 99 cent EA games, they are great and super cheap for next few days
Little things - one of the best games out there
Crosswords - expensive but ever expanding collection of real newspaper crosswords from around the country
Plants vs Zombies - a classic
Zombieville - one of my absolute favorites
Lets Golf 2 - heard nothing but good things, hoping to play with my family over holidays
there are a million more games but these are my favs.
That's about everything that I use on a regular basis, I take great pride in the apps I have found as I did a lot of research and I think you will like any of those I listed as long as you have a use for them.