Production support personal provide assistance for systems in production. For example, if a company has a web application that's deployed and available to the public and something goes wrong, the production support staff figures out what's wrong, and may provide a fix as well. It might also entail re-running back-end jobs that failed (for example in a banking position the bank might get files to process throughout the day).
It really depends on the nature of the applications/processes in place, but basically production support is the front line of defense when things go wrong.
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