Answer:
Generally speaking, you would find conductor ampacity (the amount of current it can handle continuously) in Table 310.16 of the NEC (National Electrical Code). The numbers in the table are good as long as you don't go over the stated number of conductors at the top of the table and the 86 degree Celsius ambient temperature. If anything changes, use the adjustment factors for conduit fill and ambient temp.
However, with all the possible other applicable sections of the NEC, it really depends on the type of load and how and where the cable is fixed because its operating temperature will depend on whether it is enclosed in a duct, is buried underground or is outside in the open air.
For a typical load, not knowing anything else about it, I would multiply the load (in amps) by 1.25 and find a wire with at least that ampacity.
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As always, if you are in doubt about what to do, the best advice anyone should give you is to call a licensed electrician to advise what work is needed.
Before you do any work yourself,
on electrical circuits, equipment or appliances,
always use a test meter to ensure the circuit is, in fact, de-energized.
IF YOU ARE NOT ALREADY SURE YOU CAN DO THIS JOB
SAFELY AND COMPETENTLY
REFER THIS WORK TO QUALIFIED PROFESSIONALS.