What files or directories of a release are the bare minimum to run a release?

If you do a default build, the target will be inside your _build directory, with sub-directories for the config environment and your application, e.g. _build/dev/rel/your_app/. That directory should contain everything you need to run your app -- the prompt after running mix release provides some clues for this when it says something like:

Release created at _build/dev/rel/your_app!

I find it more useful, however, to zip up the app into a single portable file (and yes, I agree that the details about how to do this are not necessarily the first things you see when reading about Elixir releases). The trick is to customize your mix.exs by fleshing out the releases option -- this is usually done via a dedicated private function but the organization of how you supply the options is up to you.

What I find is often useful is the generation of a single zipped .tar.gz file. This can be accomplished by specifying the include_executables_for option along with steps. It looks something like this:

# mix.exs
defmodule YourApp.MixProject do
  use Mix.Project

  def project do
    [
      # ...
      releases: releases()
      # ...
    ]
  end

  defp releases do
    [
      my_app: [
        include_executables_for: [:unix],
        steps: [:assemble, :tar]
      ]
    ]
  end

When you configure your application this way, running mix release will generate a nice portable file containing your app with everything it needs. Unzipping this file is education for understanding everything your app needs. By default this file will be created at a location like _build/dev/yourapp-1.0.0.tar.gz. You can configure the build path by specifying a path for your app. See Mix.Release for more options.

When you run mix release, it bundles all of your Elixir/Erlang dependencies for the MIX_ENV in question into the release directory, the erlang BEAM runtime/VM that you were using in your build, and any files that you specify in your mix project in mix.exs.

Because the BEAM runtime and code that bootstraps loading your code are included in the release, you won't need to install Elixir or Erlang on the target machine.

Things that are not included include:

• any non-Elixir dependencies. For example, if you rely on openssl, you'll need to make sure you have a binary-compatible version of that installed on the machine you plan to run on (typically, the equivalent major verson release).
• Portable bytecode. BEAM isn't like the Java VM. The compiled BEAM code needs to run on a substantially similar architecture. Build on an Arm64 machine for deployment on an Arm64 virtual machine, or x86 for Intel-compatible hardware, for instance. And it's probably best to use the same major OS distribution. There may be cases where "Any Linux * Same CPU architecture" is fine, but for example, building on a Windows or MacOS install of Elixir/OTP and deploying on Linux is a non-starter; you'd need to use a sufficiently similar OS.

As an example, one of my projects has its releases built on Alpine using Docker, so we only really have to worry about CPU compatibility. In our case we do need to make sure some external non-Elixir dependencies our app binds to are included on the docker image.

RUN apk add --no-cache libstdc++ openssl ncurses-libs wkhtmltopdf xvfb \
  fontconfig \
  freetype \
  ttf-dejavu

(ignore the fact that wkhtmltopdf is kind of deprecated, we're working on it. But for now it's a non-elixir dependency we rely on).

If you're building for a, say, an EC2 instance and not using Docker, you'd just need to make sure your release is built on a similar OS to what you're using for production, and make sure the production AMI (image) has those non-Elixir dependencies on it, or will at the time of deployment, perhaps using apt or another package manager. For a VPS, the solution for non-elixir dependencies will depend on whether they have the option for customizing the base machine image (maybe with Packer or Ansible)

Since you may seem to have been a bit confused about it in the comments, yes, MIX_ENV=prod mix release will build all of your production Elixir/Erlang dependencies and include them in the /_build/prod folder.

I include the whole ./prod folder in our release, but it looks like protocol consolidation binaries and the lib folder .Beam files are all in the rel folder so that's a bit unnecessary.