Over the years, I've come across several applications, both desktop and mobile, which I found to be beneficial when it comes to my photography.
When I want to know where the sun or moon are going to be at a particular time, I use The Photographer's Ephemeris (iOS|Android|Desktop). It helps plan outdoor photography shoots in natural light, particularly landscape and urban scenes. It’s a map-centric sun and moon calculator which helps me see how the light will fall on the land, day or night, for any location on earth.
Another app I find indispensable is PhotoPills (iOS|Android). While it has a lot of the same functionality as TPE, it has some features I find useful. It has a depth-of-field (DoF) calculator, where you enter the camera, focal length, aperture, distance to the subject and whether you're using a teleconverter; and it tells you the hyperfocal distance, DoF near/far limits, and total DoF. It also takes advantage of augmented reality and uses your phone's camera to help show you the positions and paths of the sun, moon and the Milky Way.
The next app I use, which does have limited functionality but I still find useful, is the Lee Stopper app. I use the Lee Little Stopper (6-stop) and Big Stopper (10-stop) neutral density filters when I want long exposures. This app lets me enter the shutter speed for a regular exposure and then gives me to corresponding shutter speed for the Stopper I want to use. It also has a timer so if I need more than 30 seconds and have to put my camera into bulb mode, it will let me know when to close the shutter.
I use several weather apps, mostly because I've found that while one app will report there won't be rain, another app may report rain, and I'd rather be prepared. For regular weather forecasts, I use the built-in weather app on my iPhone, AccuWeather, The Weather Channel's app, Yahoo! Weather, WeatherBug and Weather Underground. For cloud cover and wind movement, I use MyRadar. For trying to capture lightning strikes, I use an app called Storm.