If hot comes out when the handle is on the cold side, either the cartridge was installed 180 degrees off during a repair, or the supply lines under the sink are crossed. Both take ten minutes to fix and need no new parts. Pick your brand for the steps.
Pick your brand
MoenDeltaKohlerPfisterAmerican StandardGroheGlacier BayPeerless
Not sure of the brand? See how to find your faucet brand and model. New to this? Start here.
If the repair does not fix it
A new cartridge fixes most of these. When it does not, or when the faucet body itself is corroded, cracked, or so scaled that parts will not come out, replacing the faucet is usually cheaper than a second visit to the problem. A mid-range kitchen faucet from the brands covered here runs about $150 to $300 and a bathroom faucet $80 to $200, and swapping one takes about an hour with the same tools. Compare the cost of the part plus your time against a new faucet before buying the part twice.
Kitchen and bathroom faucets at Home Depot. The brand pages list each manufacturer’s own store as well.