London Tube app

I tried to use mine but found I didn't need it. The letters were too small for me to read well. I found a map online of a photo of the system and that is what I used. Sorry, I don't remember the site I used. Maybe the auto club might have a map of the system?
You will find the system easy to use. We changed our destinations frequently and a quick look at my photo or the map at any subway station got us to where we wanted to go. We had no issues.

Are you wanting an app for real-time information for Tube trains, or buses, or more for directions?

I just use the free paper Tube maps most of the time. No charging required!

I see Citymapper gets recommended quite often for London.

I use Tube Map by Mapway Limited. The free version is good enough. Works offline to tell you the best route, including when it's faster to walk between two stations.

Citymapper is quite handy. I used it in London last spring.

What I liked best is you tell it where you want to go and it tells you how to get there... by varying means -- walk. walk & tube. Walk & bus. Walk, tube, bus, walk. Whatever the mean. and it tells you which tube train you're trying to catch (the time, that is, as well as how long til the next one) and suggests were to stand to load (for easiest access) and which door to leave the train to set you up for your transfer within the station or for your exit.
What I don't recall is whether it works without using data. I spring for a 12.50GBP sim card with 1GB of data, so I didn't worry about going online or offline.
I could have used it in Paris, as well, on the same trip, but didn't realize it was designed for several cities until I have already finished that part of my trip.

I'll second Citymapper, I found it invaluable for getting around this past fall.

As to the data issue, you do need data to load up your destination and route options, but it will continue to navigate you (quite accurately) after you disconnect.

The official journey planner is on the TFL website: https://tfl.gov.uk/
The official tube map is here: http://content.tfl.gov.uk/standard-tube-map.pdf
Exactly the same map as posted at every tube station. You can print it out, copy the file to your phone (no app needed), or just pick up a free copy at any tube station. The free copy is a small foldable leaflet that fits in your pocket.
There is a standing joke that you can always spot a tourist as they are the ones staring into a tube map.

The operator of the tube is TFL, so use their website, as this will be the most up to date.

I'm going to confirm what Emma stated.

Every Londoner I know uses Citymapper. And I do as well when I'm in London.

I use Citymapper almost daily, its all you need. It also has an 'offline' tube map which is very handy.

CityMapper does the same in Paris (uses real-time data to provide information) -- that's what makes it so invaluable. It's like a miracle of the best things that connectivity can do for us IMHO.