Can energy exist without matter? Is it theoretically possible that energy will be in space without any matter/antimatter/etc in that space? Is it theoretically possible that space will exist with only energy in it and without anything else in that space?
 A: I see there's already an accepted answer, but I wanted to contribute a simpler explanation, since it sounds like that's what you're going for.
In basic mechanical physics, you're probably used to the idea of mechanical energy. When you pull a roller coaster up to the crest of the hill, you've imparted potential energy to it, which it then turns into kinetic energy as it rolls down the hill and through the ride.
But that is only one form of energy. The radio waves that you pick up on your FM radio also have energy. Unlike waves moving through water, these radio waves can travel through empty space and do not require matter to transmit them. Radio waves, light, and other forms of radiation all have energy, but do not need matter.
So yes, you absolutely can have energy without matter, in empty space.
A: Yes.  Matter is defined by the Pauli Exclusion Principle, which states (oversimplified, thank you minutephysics) that two particles of matter cannot exist in the same quantum state.  This principle applies to quarks (which make up, inter alia, protons and neutrons) and leptons (i.e. electrons and neutrinos).  However, there are other particles (bosons) which do not follow the Pauli Exclusion Principle and therefore are not matter, but still contain energy.  There is no reason that these non-matter particles cannot exist without matter particles existing.
In fact, in the early universe, the energy in space was dense enough to create matter-antimatter pairs from the pure light that was in the universe.  For some yet-to-be understood reason, this symmetry was broken and we ended up with more matter than antimatter, which is why there is still matter in the universe.  Had this symmetry not been broken, there would be no matter in the universe and only light (which has energy).
Credit hardcore to Henry of minutephysics.
