Backdoors are server-side malicious scripts which are intended to perpetrate malicious acccess to the server. The typical example of such backdoors are various File Managers, Web Shells, tools for bypassing admin login or various one-purpose scripts allowing the attacker to upload and run another type of malicious scripts. The payload is PHP based, thus intended for server-side use and the payload is executed directly on the server, while the site is loaded. Only the payload result (such as Web Shell environment) is visible in the browser, not the malicious code itself. It's very common, that backdoors don't have any visible signs in the site code and it's impossible to detect them by accessing the infected site from outside. Server level analysis is necessary in case of infection by this type of malware.
Most knew webshells are c99 and r57
Any PHP based web site (often through outdated WordPress, Joomla, osCommerce, Magento, Drupal and stolen passwords).
Cleanup is done by deleting the malicious code from the file, or replacing it with a fresh version. Reviewing access logs for non-expected HTTP POSTs can point out the possible infected files.