Below are my favorite apps broken out by category. In most cases the apps will be compatible with both the iPhone and iPad, but if it is compatible with only one I will call it out. Starred apps are among my favorite and most highly recommended.
Photographer Photo Apps
ePhotobook White Sands HD
William Neill’s Yosemite Volume One
Social Media
Twitter 
Delicious Bookmarks Pro
Facebook
Flipboard (iPad)
WordPress
PicPosterous – Posterous
Tumblr
Flickr – Yahoo!
Instagram
Best Camera
Foursquare
Plancast!
Photography Research / Reference Tools
Easy Release
iFotoGuide – Arches National Park
iFotoGuide – Yosemite Valley
iFotoGuide Grand Canyon
iBird Explorer PRO
iBird Pro HD (iPad)
Rick Sammon’s 24/7 Photo Buffet
Rick Sammon’s 24/7 Photo Buffet for iPad (iPad)
Fotopedia Heritage
Flickr Photo Map
Starmap HD
Moon Globe HD
Focalware
The Photographer’s Ephemeris
Tide Graph
Google Earth
PhotoCalc
MagicHour
DoF Calculator
Photography Editing Tools
Photogene
Photogene for iPad
Photo fx – The Tiffen Company
Photo fx Ultra (iPad)
Cool fx
Darkroom Pro
TiltShift Generator
TiltShift Generator for iPad (iPad)
Filterstorm (iPad)
CameraBag
Adobe Photoshop Express
You Gotta See This! (requires iPhone 4)
Pro HDR
Pano
360 Panorama (iPhone 4)
amopic for iPad (iPad)
iPhone Camera Enhancement Apps
ReelMoments (timelapse for iPhone)
Gorillacam
DSLR Camera Remote Professional Edition
Darkroom Pro
Night Photography
GoSkyWatch (iPhone)
GoSkyWatch (iPad)
Moon Seeker
Starmap (iPhone)
Starmap HD (iPad)
Satellite Visibility (iPhone)
ISS Visibility HD (iPad)
ISS Visibility (iPhone)
ISS Visibility HD (iPad)
Penki
Aurora Forecast
The Photographer’s Ephemeris (iPhone)
The Photographer’s Ephemeris (iPad)
Focalware
Weather
The Weather Channel® Max
The Weather Channel Max for iPad (iPad)
Weather Alarm Clock
Utilities
1Password Pro
Photo Transfer App
Skype (works on iPad also!)
Dropbox
Instapaper
Evernote
Things (iPhone)
Things for iPad (iPad)
Bump
Speedtest.net Mobile Speed Test
QuickMark QR Code Reader
QuickMark QR Code Reader 4 (iPhone 4)
Convertbot — The Amazing Unit Converter
CAMERA-A (iPad – displays video via CAMERA-B from your iPhone)
CAMERA-B (iPhone)
White Noise Pro (iPad)
Find My iPhone (iPad – MobileMe Required)
Business
Keynote (iPad)
Keynote Remote – Apple Inc. (iPhone)
FolioBook
Genius Scan
San Francisco Parking
Square
PayPal
Yelp
Ustream Live Broadcaster
Navigation
MotionX GPS Drive HD (iPad)
MotionX GPS Drive (iPhone)
Entertainment & News
The Guardian Eyewitness
NPR News (iPhone)
NPR for iPad (iPad)
BBC News
Pandora Radio.
Shazam
Shazam for iPad (iPad)
IMDb Movies & TV
Wikipanion Plus for iPad
Pulse News Reader (iPad)
GoodReader for iPad (iPad)
GoodReader for iPhone (iPhone)
NASA App HD – NASA (iPad)
Art
Da Vinci HD (iPad)
Van Gogh HD (iPad)
Monet HD (iPad)
Michelangelo HD (iPad)
Art Authority for iPad. (iPad)
Games
Words With Friends (iPhone)
Words With Friends HD (iPad)
Doodle Bomb: Physics Puzzle with a Bang (iPhone)