Weakness very rarely occurs just on it's own, so the chances are that you're experiencing some kind of pain as well and possibly even some tingling or numbness.
What does a feeling of weakness mean?
Most often we see weakness in association with pain. So for example, you feel like you can't lift your arm up as easily as before, and you have shoulder pain. That weakness is probably more due to the fact that it hurts to make an effort than actually being weak. However, we use muscle testing in some pain cases to help identify where a problems stems from. For example, you have neck pain, and we see how strong the different muscles in your arm are and your biceps are noticeably weaker on one side than the other, this points us to a specific nerve and a specific part of your neck.
Is it serious?
Weakness as a symptom is generally more serious than tingling, numbness or pain. This is because the sensory parts of our nerves are much more...well, sensitive than the motor parts. So when you start to notice weakness it usually means there is a higher degree of nerve interference than some of the other symptoms.
The take home message with this symptom is that if you've started to get it, it's time to get checked out by a professional.