What are Salesforce programming languages?

Backend

You cannot use other programming languages for backend development than Apex, but have in mind that within Salesforce you can use a great number of APIs, from which the most popular ones for developing custom applications are REST API and SOAP API. For mobile development you can use Mobile SDK.

May 2022 update:

Recently, Salesforce has launched a new product that supports writing code in Java and JS (Node) - Salesforce Functions. Here's a description from the home page:

    Salesforce Functions lets you extend the Customer 360 Platform with custom elastically scalable business logic. Salesforce Functions is designed to boost developer productivity by reducing your infrastructure responsibilities and enabling you to build and integrate Functions-as-a-Service (FaaS) apps using the languages and tools of your choice.

Some useful links in this area:
Get Started with Salesforce Functions
Salesforce Functions Developer Guide
Explore Solutions and Sample Code
Frontend

Also, Salesforce frontend development is basically JavaScript. The newest framework - Lightning Web Components Framework uses W3C Web Components Standard. Apps written in LWC can perform basic database operations.

You can also host apps written in other JS frameworks what's nicely explained in this article: Lightning Container Component: Building Components with React, Angular, and Other Libraries