Is dark energy (like) normal energy?

What we know is that the mass-energy density of the universe must be much higher than expected from baryonic matter/dark matter in order to fit the accelerated expansion observed. It could be a particle. It could some gravitational feature we don't understand... The thing is, what we currently think is that the density of dark energy is constant over time. That's pretty crazy by itself, but it's even crazier if you try to figure out how to do that with some kind of particle.

The most reasonable way to get a density that doesn't change over time, is to say that it's a property of space; if you get more space, you get more "space energy". Dark matter is likely to be a particle, since it "clumps" around galaxies. But dark energy seems to be basically everywhere.

When we say 'dark energy', are we saying that it's a different concept altogether than energy?

Yes, it is. Thermodynamics fundamental equation states that when gas volume increases,- then gas internal energy should drop, assuming it has constant entropy:

$$ mathrm {d} U=-p,mathrm {d} V $$

But contrary to expectations, universe does not slows-down in expansion, but even accelerates the expansion with increasing universe volume. This means that in given thermodynamics equation above pressure term is negative, that's why universe has overall increasing "internal energy".

How pressure can be negative ? Pressure can be expressed as vacuum energy per unit volume : $$ p=frac {E}{V} $$

Substitute Einstein's mass-energy relation (noticing that multiplying both equation sides by $-1$ does not change anything in principle mathematically, however physical meaning will be different) $$ -E=-mc^2$$ So we get for pressure :

$$ p=frac {-mc^2}{V} $$

This equation can be a bit shortened, due to the fact that it involves vacuum density :

$$boxed {p= rho_{text{vac}} c^2 }$$

Currently vacuum density is estimated to be : $$ rho _{text{vac}}=-5.96times 10^{-27}{text{ kg/m}}^{3} $$

Negative vacuum density lets for universe to "accumulate" internal energy. It needs some sort of negative mass, which fills all vacuum space. Negative mass gives birth to "negative gravitation", i.e. negative mass is repulsion force, not attraction like gravity. So two units of negative mass will propel from each other apart. That's why "dark energy" is called dark, it can be thought as a synonym to a negative energy.