One simplest candidate for dark energy is a cosmological constant.
Is cosmological constant some kind of field that permeates all of space, or is it just some kind of energy stored in space?
@Forge Cosmological constant is not a candidate for dark energy, it is the quantity which denotes dark energy.
And yes it is believed that dark energy fills up all of spacetime but the theoretical calculation of vacuum energy density vs Experimental measured value have largest disagreement ever with respect to any disagreement in science. Theoretical calculations predicts about 10^(108) order of discrepancy. This is called Vacuum Catastrophe.
Also it's more preferred to look at dark energy as a antigravity force field without any dark energy particle content. But not a single one interpretation is more accurate than other because we don't yet know what dark energy force field is or even if it made of dark energy-particles or not.
The cosmological constant has the same effect as an intrinsic energy density. Although it was introduced as a constant, and intended to be positive, you can build the theory with a negative cosmological constant, or one that varies in time in some preferred frame (and so is not constant) or one that varies in space and time the right way and as such becomes a scalar density field. All of these variants are mathematically natural, but evidence for it is controversial.
Although I feel that @Aman_Pawar's comment that it is the quantity that denotes dark energy is misleading in being a bit incomplete - the point is that it is not a separate entity, such as some kind of quantum particle, that would, when combined with relativity require inclusion as an adjustment to the cosmological constant, it is literally the way in which the energy density is expressed in Einstein's equations.
I expanded on this because of the part of your question in which you asked - is it some kind of field. And the answer to that is that it would be mathematically natural and make physical sense for it to be interpreted as a measure of some density. The next question is whether this field is observed empirically.