The Priceline Negotiator on iPad is on my fun list when it comes to travel. Its a great touch experience with realtime updates of retail and name your own price hotels. It has a fun design and a streamlined checkout. Delta has stepped up their game with their iPhone app. Its efficient and handles data caching well for offline viewing.
I'm a big fan of Foodspotting for finding food in an unfamiliar city. It helps find good food near you, similar to Yelp but for specific dishes rather than restaurants.
For example, if I get to my hotel and want to see what good dishes are nearby, it brings up pictures (always tasty-looking) of what people have recommended with the restaurant name, a map, and distance away. Alternately, I can type in a dish (ex. hamburger, bibimbap, etc.) and it will bring up photos with distances and restaurants that are the nearest to me. When I am in a new place and have a hankering for something specific, it's ideal. http://www.foodspotting.com
For booking flights, Kayak does a nice job of aggregating all the available options and linking you through to the associated carrier for purchase.
Airbnb is the first place I look when considering where to stay, but am curious to test out Hotel Tonight for last minute deals on reservations.
In the air, Instapaper is fantastic for its offline reading feature. Just make sure you have a backlog of unread articles to keep you in content.
And once I'm on the ground, Yelp is still my go to for finding the best eating options for wherever I may be.
My Next Trip uses foursquare and plans an itinerary for you based on your destination and foursquare chatter.
Tripit / AMEX Travel app is great. If you have the AMEX Platinum they give you Tripit Pro as part of your membership. This gets you a customized version of the Tripit app. If you've loaded your travel data into Tripit, the app gives you offline access to all the locations and schedule for your trip. Pretty handy if you don't have a data connection when you get to another country.
The British Airways iphone app is a must if you fly with BA, it allows you to check in using your phone and sends a QR code to your phone for check-in at airport security and boarding. Even the status of your flight is often more up to date on the app than the flight status shown at the airport.
The lonely planet city guide apps are excellent and always download them depending on where I am going. Often you can get them for free from the app store. For a bit of luxury and fun the Mr and Mrs Smith app will cheer anyone up http://blog.mrandmrssmith.com/2011/07/hotel-iphone-travelapp/.
GoHow Airport - Consistently provides the best realtime flight information (gate changes, delays, incoming flight times, etc) as well as great airport amenity and shop info, and it's free. The interface isn't the best, and the app crashes a bit, but unlike most other apps of its kind, it has alerts, so I don't have to constantly be checking my phone. Kayak Flight Status uses the same info, but doesn't do alerts.
Airbnb. (I guess everyone will answer that)
Google map. (a must have)
In Europe, I use voyage-SNCF (ticket booking, very efficient), AirFrance (mileage and services) and airports' apps when they have one (like AEROPORTS de Paris), in order to get the local up to date information.
And when I go and visit other countries, for leisure or for work, I download the 10 first apps I'll find in the AppStore related to the destination. And I do this afr checking in, waiting for my plane. Most of the time, I'll try to download a " local tourist info" app, a "metro map" app and a "where to go go out at night" app.