If you need the certificate against a deadline, say so in the subject line and the first sentence, give the exact date you need it by, and offer to collect it in person or accept a scanned copy first. Deadlines that are stated get prioritised; deadlines that are implied do not.
An experience letter confirms what you did and for how long. A relieving letter confirms that you completed your notice period and were formally released. New employers in India and the Gulf frequently ask for both.
Practice varies by jurisdiction. Many employers issue one as a matter of policy, and some labour laws require a service certificate on request. If they refuse, offer letters, payslips and tax records are usually accepted as alternative proof.
HR or the people operations team, with your reporting manager copied. Managers rarely have the authority to issue letterhead documents themselves.