Backend services that need high reliability, low latency, lots of concurrency. Lower p99 latency and better concurrency model than Java, more reliable/productive than C++.
I'm at Tinfoil Security, and we use Elixir for dynamic security scanning of web applications and APIs. We also have some Phoenix apps, but the bulk of our Elixir use isn't in web development.
It's a very IO-bound problem (how many requests I can make while staying within rate limits / not taking down a service), so Elixir is a great fit for coordinating all of the concurrent work involved in security scanning.
Other uses where Erlang shines are a great fit for Elixir too: WhatsApp, Goldman Sachs uses Erlang for Trading, Heroku's routing mesh, ...
I think with Nerves, Elixir fits well on embedded platforms. Its supervision trees make it easier to create fault tolerant systems, and the actor model is great for concurrency.
I use it for everything from bots to interacting with and controlling Java and Python programs and a lot more. The concurrency abilities of the BEAM have a lot of applications.