There's a big difference between discomfort and pain during a workout. Discomfort normally goes away when the exercise is completed. Pain lingers. When the pain is serious enough, your body compensates by using the surrounding muscles which can further complicate things. It's important to pay attention and listen to your body. If you take care of the pain early on, there's a good chance it won't progress into an injury.
Common Workout Pains
Sharp/Pinch: This kind of pain happens when there is inflammation of muscles or tendons. The "pinching" is typically felt near or at a joint (for example, the shoulders or hips).
Pain with Swelling: Swelling is caused by inflammation. If swelling persists, there might be an injury that needs to be looked at by a medical professional.
Tingling: This is normally a sign that nerves are involved. When their pathway is inhibited, the surrounding nerves will send signals to the brain (tingling sensation) that something is wrong.
Worsening/Long-lasting: Pain that either worsens throughout a workout or while resting is a sign that there is a further injury, and should probably be checked by a doctor.
Painful "Pops": General "pops" isn't grounds for concern, but a "pop" followed by pain can be a sign of a tear or dislocation.
What to do When You Feel Pain
Check Your Technique: First, make sure you're going through the motions of the workout correctly.
Go Slow: There's no need to rush through your workout. Rushing can hinder your form. Sometimes lowering the intensity of the workout can subside the pain.
Rest: Taking time off can allow your body to heal. If pain persists, even with rest, there could be a deeper injury.
See a Doctor: If the pain you're experiencing doesn't go away, seeing a doctor is a good idea. This assures that there is or isn't an injury. A doctor could also give you stretches or exercises that can strengthen the area where the pain originated.
What NOT to do When You Feel Pain
Don't Overdo: Working out longer or more intensely can create more pain. If you're already feeling pain, don't go harder.
Don't skip a warm-up: Warming up (stretching or walking) your muscles before a workout can help prevent injuries while working out.
Don't Ignore the Pain: Pain is the body's way of telling you something is wrong. If whatever pain you're experiencing doesn't go away, worsens, limits mobility, swells, or bruises, you should see a doctor.