NPM/Bower/Composer - differences?

Question

[update, four years later]

bower is deprecated, and should not be used anymore for new projects. To a large extent, it has been subsumed into node dependency management (from their website: "While Bower is maintained, we recommend using Yarn and Webpack or Parcel for front-end projects").
yarn came out of the wood as a better npm (fixing several of npm flaws), and this is really what you should use now, as it is the new de-facto standard if you are doing front-end or node development. It does consume the same package.json as npm, and is almost entirely compatible with it.
I wouldn't use composer at this point (because I wouldn't use php), although it seems to still be alive and popular

[original answer]

npm is nodejs package manager. It therefore targets nodejs environments, which usually means server-side nodejs projects or command-line projects (bower itself is a npm package). If you are going to do anything with nodejs, then you are going to use npm.

bower is a package manager that aims at (front-end) web projects. You need npm and nodejs to install bower and to execute it, though bower packages are not meant specifically for nodejs, but rather for the "browser" environment.

composer is a dependency manager that targets php projects. If you are doing something with symfony (or plain old php), this is likely the way to go

Summing it up:

doing node? you do npm
doing php? try composer
front-end javascript? try bower

And yes, the "json" files describe basic package information and dependencies. And yes, they are needed.

Now, what about the READMEs? :-)