The list of most common applications for Erlang as been covered (CouchDb, ejabberd, RabbitMQ etc) but I would like to contribute the following.
The reason why it is used in these applications comes from the core strength of Erlang: managing application availability.
Erlang was built from ground up for the telco environment requiring that systems meet at least 5x9's availability (99.999% yearly up-time). This figure doesn't leave much room for downtime during a year! For this reason primarily, Erlang comes loaded with the following features (non-exhaustive):
Horizontal scalability (ability to distribute jobs across machine boundaries easily through seamless intra & inter machine communications). The built-in database (Mnesia) is also distributed by nature.
Vertical scalability (ability to distribute jobs across processing resources on the same machine): SMP is handled natively.
Code Hot-Swapping: the ability to update/upgrade code live during operations
Asynchronous: the real world is async so Erlang was built to account for this basic nature. One feature that contributes to this requirement: Erlang's "free" processes (>32000 can run concurrently).
Supervision: many different strategies for process supervision with restart strategies, thresholds etc. Helps recover from corner-cases/overloading more easily whilst still maintaining traces of the problems for later trouble-shooting, post-mortem analysis etc.
Resource Management: scheduling strategies, resource monitoring etc. Note that the default process scheduler operates with O(1) scaling.
Live debugging: the ability to "log" into live nodes at will helps trouble-shooting activities. Debugging can be undertaken live with full access to any process' running state. Also the built-in error reporting tools are very useful (but sometimes somewhat awkward to use).
Of course I could talk about its functional roots but this aspect is somewhat orthogonal to the main goal (high availability). The main component of the functional nature which contributes generously to the target goal is, IMO: "share nothing". This characteristic helps contain "side effects" and reduce the need for costly synchronization mechanisms.
I guess all these characteristics help extending a case for using Erlang in business critical applications.
One thing Erlang isn't really good at: processing big blocks of data.
