How to find out what programming language a website is built in?

There's no way to be 100% sure if you don't have access to the server, so it's about guessing. Here are some clues:

• File extensions: login.php is most likely a PHP script.
• HTTP headers: they may leak some information about the language which is running on the server, and some additionnal details like the version: X-Powered-By: PHP/7.0.0 means that the page was rendered by PHP.
• HTTP Parameter Pollution: if you managed to guess which server is running, you can refine the guess.
• Language limits: maximum post data, maximum number variable in GET and POST data, etc. If may be useful if the webmaster kept the default values.
• Specific input: for example, PHP had some Easter eggs.
• Errors: triggering errors may also leak the language. Warning: Division by zero in /var/www/html/index.php on line 3 is PHP, for example.
• File uploads: libraries may add metadata if the file is being modified server-side. For example, most sites resize users' avatars, and checking for EXIF data will leak CREATOR: gd-jpeg v1.0 (using IJG JPEG v90), default quality, which may help to guess which language is used.
• Default filenames: Check if / and /index.php are the same page.
• Exploits: reading a backup file, or executing arbitrary code on the server.
• Open-Source: maybe the website has been open-sourced and is available somewhere on Internet.
• About page: the webmaster may have thanked the language community in a "FAQ" or "About" page.
• Jobs page: the development team may be recruiting, and they may have detailed the technologies they're using.
• Social Engineering: ask the webmaster!
• Public profiles: if you know who is working on the websites, you can check their public repos or their skills on online profiles (GitHub, LinkedIn, Twitter, ...).

You may also want to know if the website is built with a framework or a CMS, since this will give information about the language used:

• URLs: directories and pages are specific to certain CMS. For example, if some resources are located in the /wp-content/ directory, it means that WordPress have been used.
• Session cookies: name and format.
• CSRF tokens: name and format.
• Rendered HTML: for example: meta tags order, comments.