When a creature would be affected by something they are resistant to, they make a resistance save. This is a DC 10 d20 roll that they add their proficiency bonus to if they are proficient in the ability that grants them their resistance. On a success, the target is not affected by whatever was going to happen to them. On a fail, they are. Resistance saves cannot be failed automatically. Only one resistance save can be attempted per interaction. A resistance cannot stop a warded ability.
If an item is resistant to something (such as if it is durable) and it is being used, then the save is made with its user's proficiency bonus. Otherwise, it is made without a proficiency bonus.