A strain happens when you stretch a muscle too much, as in "I strained my quad or bicep".
A sprain is an actual injury to a muscle AND the ligaments, tendons, or soft tissues that surround a joint, as in "I sprained my ankle".
Normally, you need to see a doctor to determine if an injury is a strain or sprain, typically with an x-ray or an MRI. Generally, a sprain is a little more severe than a strain, although they can both be pretty painful.