In kabbalistic teachings, the number seven symbolizes perfection – perfection that is achievable via natural means – while eight symbolizes that which is beyond nature and its (inherently limited) perfection.
a) G‑d created the world and its natural order in seven days.
b) Between the holidays of Passover and Shavuot we count seven weeks. During these weeks we work on perfecting our seven emotions (love, fear, compassion, ambition, humility, bonding and receptiveness)—one emotion per week. At the conclusion of these seven weeks we have sufficiently perfected ourselves to be worthy of receiving the Torah on Shavuot. A completed person has control over all seven emotions.
c) The Holy Temple's menorah, which served to illuminate the natural world with the holy glow of spirituality, had seven branches.
e) There are seven colors of the rainbow and seven musical notes. When something has seven parts, it symbolizes that that it has reached its state of completion: the seven notes of the diatonic scale make one complete octave, etc.
Eight, on the other hand, is symbolic of an entity that is one step above the natural order, higher than nature and its limitations. That's why Chanukah is eight days long—the greatly outnumbered Maccabees' resolve to battle the Greeks wasn't logical or natural. They drew on reservoirs of faith and courage that are not part of normative human nature. They therefore merited a miracle higher than nature – a miracle that lasted eight days – and to commemorate this, we light on Chanukah an eight-branched menorah (click here for the full story). In a similar vein, we circumcise our children when they are eight-day old babies, because the brit milah symbolizes our nation's supernatural and logic-defying covenant with G‑d.