Long answer:
If you recall the equipartition theorem, it states that a system has an average energy of kT/2 in each quadratic degree of freedom (mode in which the gas can store the energy). A monatomic gas can store energy as kinetic energy (Vx^2, Vy^2, Vz^2), so the average energy of an atom in this gas is 3*(kT/2), and the total energy of the gas is just the number of atoms times the average energy: U = N*3*(kT/2).
A diatomic gas can store the energy in the three translational degrees of freedom and two additional modes of rotation, so they have 5 degrees of freedom. Hence the average energy of a diatomic gas is 5*(kT/2), and the energy of the total gas is U = N*5*(kT/2).
The nuance is that if a particular way of storing energy isn't expressable as a square of some variable in the expression for total energy, the theorem doesn't hold, and it may not have an average energy of kT/2. For example, the total energy of a free particle moving in three dimensions is:
U = T = Px^2/(2m) + Py^2/(2m) + Pz^2/(2m) ; where the Ps are momenta.
Each term is a square of something, so under the theorem each will have an average energy of kT/2. Thus if you're given the temperature of a gas of known composition, you could figure out the average velocity or momentum in some direction after looking up the mass on a periodic table and plugging in the given temperature. My brain's shot right now (damn micro), so I'm having trouble thinking of a nonquadratic energy that isn't a potential, and I don't think the makers of the MCAT care.
Short answer:
A degree of freedom is a way the gas can store energy that can be expressed as expression involving the square of some variable, and the average energy in this degree of freedom is simply kT/2.
