Determine if the music taken from the original source was transformed with the intent of adding new expression or meaning; if so, you're most likely safe. If you're simply copying the music without adding anything to its expression, then you're in violation of copyright law. The exact definition of added meaning is, along with the other guidelines for fair use, a debatable concept. The Stanford University's Copyright and Fair Use Center has compiled an extensive database of previous court cases for public review --- if you're not sure whether your use of the music constitutes new, personalized expression, then consult past court rulings to gauge previous examples of proper and improper expression.
Consider the nature of the copyrighted work. Stanford University notes that previously published pieces are more likely to be okay for educational, personal, or transformational use, whereas unpublished pieces are not.
Consider the length of the piece both independently and in relation to the whole. Generally speaking, the shorter the clip the better. Also, make sure never to use the bulk of a clip, only take a portion of the entire work --- even when that work is already very short.
Factor in your usage's effect on potential sales. This is by far the most important factor to consider, since you are not allowed to use copyrighted work if doing so will deprive the original musician of potential income. For limited or entirely educational usage, you're usually allowed to use copyrighted material without it being considered a negative impact on the artist's sales.
Consider if your video can be considered a commentary or parody of the original music piece. If so, Stanford University says you are entitled to use small portions of copyrighted music to achieve your purposes.