XE Currency -- A free ap by one of the most popular foreign currency exchanges. Allows you to monitor multiple currencies and instantly convert amounts at the most recent bank rate to a currency of your choosing. I use this multiple times a day when traveling. I normally have the currency selected for whatever country I am in and then additional currencies I like to monitor (USD, EU, NZD).
Maps.ME -- A free map program with GPS routing that doesn't require a data connection. You need to select the country/city you will be visiting in advance and download those specific maps. The maps are open source, free and can be quite detailed with restaurants, gas stations, places of interest etc. Depending upon how large a city or country is the download size can be quite large -- up to 400MB (although the entire country of Tajikistan is like 13 mb). The trip routing is extremely useful and it will give you step-by-step directions by vehicle. However there is no audio component so you need to be careful when driving as you will have to look at the screen to follow the directions. This ap tends to suck up your battery if you let it run in the background so always exit out when you are not using it. If using it in a car for directions than make sure you phone is plugged into a charger.
Google Maps -- My go to map of choice whenever I have a data connection. Has street level details and gives great directions for a variety of modes of transport. However you need to have a data connection to take advantage of all of the features (including directions). It is possible to save certain parts of the map for offline use by bringing the section you want to save up on the screen and then typing OK MAPS in the lookup field. This only works in certain parts of the world. For some reason the save feature is blocked in certain countries (Russia. I think Japan etc)
ConvertPad – A simple measurement convertor for metricphobic Americans. It has an exhaustive list of things to convert to and from. Pretty simple to use but I reccomend going through and removing some of the measurements that you don’t care about to cut down on clutter. For example, the only temperature values I use are C or F.
WorldTime -- Basic clock that lets you display multiple time zones on one screen. Great for people who keep in touch with friends and family around the globe and need to always know the time at their locations.
Flashlight -- Allows you to use your phone's camera flash as a flashlight. Great for exiting hostel dorms late at night or exploring caves/crypts.
Translate -- A freebie by Google, this ap is extremely helpful when you find yourself in a country that doesn't share your love of english. It offers a variety of features when you have a data connection: scan and translate written text, speech recognition, voice playback and basic word lookup for almost every language in the world. Without a data connection it is useless unless you have downloaded the appropriate language pack -- so plan ahead. If you are interested in a langauge that doesn't use the roman alphabet (such as cyrllic) then you will need to adjust your keyboard settings (I use Cyrllic On Demand for traveling in Russian speaking countries)
Colornote -- I wanted a simple ap that allowed me to type in quick notes when walking down the street or sitting in a cafe. I didn't want any bells and whistles or extra features and this delivers flawlessly. My one complaint is that it doesn't automatically backup notes to my google drive account.
ES File Explorer -- Not necessarily a travel specific ap but useful for maintaining the file directory on your phone. I use it a lot to backup pictures on my phone (I have an adaptor cable to allows me to plug in SD cards or USB sticks).
Kayak -- This needs no explanation. My favorite travel site for flights in convenient ap form.
AP MObile -- A news aggregator from the Associated Press. when you have an internet connection it constantly updates a varirty of topics with the most current news. Great way for staying up to date while traveling.
SumRando -- A free VPN that can allow you to access Facebook or other blocked sites when using the internet in countries that have internet restrictions. I used this, along with OpenVPN, when I was in Iran but it may no longer work.
Tripadvisor -- Probably the best overall travel ap out there. It has all of the features of the website in convenient to use ap form. If you have a data connection it will show you all of the nearby restaurants/hotels etc for an area with reviews. For many cities you are able to download this information for offline use. The more popular cities have free guides that you are able to download that aside from all of the restaurants and hotels includes most of the tourist sites to visit (such as museums etc). A map is also included. As you review all of this you can save the ones you like and then they all appear as stars on a map which I find helps with day to day sightseeing as you are able to quickly see what is in the neighborhood you are visiting.
CamScanner -- A good office program for scanning documents and saving them as PDFs. I use this for sending back deployment documents for work. It uses your phones camera to take a picture of each page you need to "scan" and then collects them all in a PDF file.
White Noise Free -- A white noise generator that also has some nature sounds. Great for drowning out snorers in a hostel dorm.
Tinder -- Could be useful for meeting people in a new city. I haven't had any luck. It is fun for virtual people watching and it is interesting to see the way photos change in different cultures.
Weather Underground -- My weather ap of choice. It can install a small widget that displays the current temperature on the main menu screen of the phone. You can load the ap to find out extended forecasts for your current location as well as other locations of your choosing.
Capitaine Train -- For some reason it is very difficult for Americans to buy train tickets in France online -- you have to use an American based company that charges a large commission. This ap allows you to buy directly from the French rail company and not pay the commision. It also is great for checking out train schedules and monitoring your upcoming trips. I used it once to buy tickets but didn't end up taking the train so I don't know if the tickets were actually purchased! :) This also works for countries neighboring France. Requires a data connection.
Skype -- Don't need to explain this. One thing I like about using Skype is that there is no charge to make a voice call to the US if you are dialing a toll free number. I find that I have to call my banks at least once when I am traveling and since they all have 800 #s there is never any charge. Plus I rarely buy local SIM cards so skype is the only phoning capability I have.
GPS Coordinates -- Simple program that displays your current GPS coordinates and elevation.
Ankidroid -- I just started using this program and already feel it is going to be good. It is a flash card program that helps you memorize any type of data you are interested in. For traveling it is useful to load up a "deck" of flash cards with key words for the country you are in. Whenever you have a few free minutes just load it up and work on memorizing words. Supports audio and pictures as well. It is scientifically designed to help you remember data by re-displaying items you arent sure of more frequently then it displays items you are confident of.
I am still trying to find a good, simple, android ap for budget/expense tracking. The only decent one I have found (TrailWallet) is only for iOs.