1) Dark energy/Dark matter. As of right now, we can only infer the existence of these "materials". We have no idea what exactly they are of why they have the effect on the universe they do.
2) The Fermi paradox. Given that the universe is roughly 13.5 billion years old, and intelligent life has only existed on Earth for ~20,000 years (depending on how you define intelligence), there ought to be intelligent races millennia, if not eons, ahead of us. Why can we find no evidence of their existence?
3) What exactly is consciousness? What is the difference between our brains and a highly sophisticated computer? Is there a difference? (Closely related: Do we really have free will?)
4)Unified Field Theory: Many, if not most, physicists are convinced that the 4 major forces (electromagnetism, gravity, strong force, weak force) will merge into one unified force under the right conditions. We haven't figured this out yet, Einstein went to his grave working on this one.
5) The origin of life. We (mostly) understand evolution and divergent species origin, but we still don't know what sparked the whole process. This is one of the few areas where you will find some scientists willing to admit the answer might be "And God Spoke."
