Black body and black body radiation
The black body is defined as an object that absorbs all the radiation incident on it - i.e., none of this radiation is reflected. This however does not mean that it does not emit any radiation - it does emit and this radiation has black body spectrum. The emitted and the absorbed energy are then equal, so that object itself remains in thermal equilibrium.
So defined black body is actually a historical artefact from pre-quantum times, which led to deriving the Planck's law. In fact, a more robust definition of black body radiation is that of a radiation in thermal equilibrium (i.e., photon gas in thermal equilibrium - aka, Boltzmann distribution). Deriving the Planck's law then becomes a simple exercise... provided that one is at ease with quantum mechnaicsmechanics and the occupation number representation.
What kind of radiation is actually emitted by bodies
Black body spectrum used to descirbedescribe objects, such as stars, human body or fire in an oven is actually an approximation. The approximation assumes that the body is (a) in thermal equilibrium, (b) that it can emit at any wave-lengths, and (c) that emitted energy is too small to significantly change the thermal state of the body. Indeed, if the body were not in thermal equilibrium, it could emit at some frequencies more than at others - e.g., a neon lamp. The same neon lamp can serve as an example of an object that emits only certain wave lengths. Finally, if the body loses a lot of energy via radiation, it would cool down and its radiation spectrum would shift in time towards lower frequencies. However, the approximation works admirably well in many cases, whereas deviations from this approximation allow, e.g., to determine the chemical composition of stars.
Equilibrium state vs. steady state
Actually, that object receives the same amount of energy as it emits is not sufficient for it to be in thermal equilibrium - i.e., its state may not be describable by Boltzmann statistics, and it may not be in equilibrium with its environment.
E.g., the Earth is heated by the Sun, which can be considered a black body at temperature of about 6000K. The Earth then emits some heat to vacuum, which can be considered as an environment at temperature 0K. The Earth is neither in thermal equilibrium with the Sun, nor with the vacuum, but the relaxation processes on Earth can be considered rather fast, and its state can be considered as an equilibrium state at some temperature intermediate between 0K and 6000K (which is gradually rising). So Earth is in a quasiequilibrium steady state.
SameThe same reasoning can be applied to a human body, see, e.g., here and here.
See also:
Does fire emit black-body radiation?
How does radiation become black-body radiation?
Black body vs. Thermal radiation
How do photons reach thermal equilibrium with the walls of the blackbody cavity?