Springpad - this one has been my favorite lately, you can create notebooks for virtually everything (sorry, Evernote) and put images, checklists (with due dates), notes, etc in them, share them with others and even choose others' public notebooks and personalize them.
TripIt - (free version) you can keep your travel itinerary, booking codes and even maps and weather forecast for your destination.
Tripomatic - You choose your destination and dates, the app gives you attractions, restaurants, sights, etc that you can add to each day, then it creates a map (also printable) with your itinerary. Good app, but it's missing many sights for lesser known destinations.
Dropbox/Drive - Life-saving cloud storage, I use it on my trips to store copies of documents and bookings, and if I have wi-fi in the hotel, then I upload the photos I've taken, just to be safe
Yelp - I've been relying on Yelp more and more lately, it has become my favorite app for reading (and also posting) reviews of almost everything. I like the "check-in" feature.
Booking - one of the best websites for hotel bookings I know. I normally book my hotels well in advance using the website but it is useful when I'm away planing my trips. They keep updating it with useful features.
Orbitz - My favorite website for hotels and vacations bookings. The app is OK but I prefer the website.
Skyscanner - like Booking, I mostly use it for travel planning, when looking for flight routes and prices.
XE Currency & Oanda Currency converters - I've been using these two to calculate currency conversions approximately, both good, I like XE's multiple currencies in the same screen, but I also like Oanda's 'choose a commision rate' option.
Unit Converter - This one converts everything you can think of: area, distance, mass, data, etc. They also have currency conversion but I prefer the above two.
Google Translate - Obviously the most useful app when traveling in a place where you don't know the language. You can either record the voice, write it down or even take a picture of the text and the app will translate it for you (the photo option is not very precise though).
Maps - it's good when you have an internet connection, otherwise you should download offline area maps (and I still haven't figured out where does the app save them in my phone!). However, it's great as a navigator too!
City Maps 2Go - This useful app downloads oflline versions of city maps.
Evernote Food - For foodies like me, it's useful to have this one when you go to a restaurant so you can write down what you ate and take a picture of it for future reference (it synchronises with your Evernote account).
Note Everything - When I'm walking around and I think of something important to to do or to remember, instead of stopping and writing (never walk and type), I just record it in this useful Notes app.
Trains Timetable IT - Best train timetable app for Italian trains (it has all Trenitalia, Trenord and now also Italo train timetables)
Italo Treno - the brand new app of my favorite Italian train company, you can buy tickets, check timetables, point balances, etc.
Reading on the go:
Kindle - I love to have something to read on planes, trains, hotel rooms, etc. so being able to bring even 10 books without adding a gram to my carry-on is one of the many things that make me happy. Kindle doesn't need an internet connection but be sure to load your books before leaving home.
I've been buying E-guides lately and I'm quite happy with them, especially because I can travel lighter and some even have links to websites, Google Maps or offline maps (Lonely Planet guides, mostly)
Pocket - Before leaving home, I download some articles on Pocket so I can read them later without using data connections. Only con, they take too much space in my phone.
Zinio - Same as Kindle but for magazines, just remember to download your magazine issues before leaving or using wi-fi.
Instagram/Instaweather/Instaplace - Because most of us like to brag about our trips to jealous friends back home :)
Whatsapp/Telegram - Instead of using Skype, I find these two better to comunicate while abroad, you can also send images, audio notes and location. (Whatsapp has a yearly-cost, Telegram is free)
Authentic weather - Probably not super accurate, but I love an honest app!
IlMeteo - very accurate for Italian weather (as accurate as that can be...)
(*all Android apps, I'm not an Apple fan, also all free apps)