How can light emit three kinds of spectra?
The atomic spectra come from electron transitions. This means that when an atom absorbs a photon its electrons go from one configuration (possibly the ground state) to a configuration with an energy that lies exactly $E_{ph}$ (energy of the photon) higher. The reverse process results in the emission of a photon. The spectrum is a line spectrum, because the electron energy levels are quantized (i.e. only certain discrete energy values are allowed) and the following equation is always fulfilled: $$E_f - E_i = \pm E_{ph}$$
Where the sign depends on whether we are talking about absorption or emission of photons.
Molecules have, with respect to atoms, more degrees of freedom. Imagine a molecule build up from two different atoms (e.g. CO). These molecules, just like single atoms, have electronic transitions. However the molecule also has vibrational and rotational transitions. The vibrational transitions come from the fact that the atoms that are part of a molecule can move with respect of one another (e.g. The C and the O atoms in CO can move towards or away from each other.).
The rotational transitions come from the fact that the molecule can rotate as well. (e.g. the C and O atoms in CO can rotate around the center of mass much like the moon and the earth rotate around theirs.) The energies associated with the vibrational and rotational transitions are also quantised. However the separation between them is much smaller than in the case of the electronic transitions. Because the transitions (and thus the photon energies) lie so close together the individual lines seem to form a band. Therefore molecular spectra are characterized by line (electronic transitions) and by band (vibrational and rotational) emission.
Up till now we have been talking about single atoms or molecules. However the concept of black body radiation is a thermodynamical concept. This means that in order for black body radiation to occur we need to be in the thermodynamical limit: We need many degrees of freedom or put in an easier way, we need a lot of atoms or molecules. The black-body radiation is a property or effect of the whole collection of atoms and molecules in equilibrium with the environment! The radiation originates from the motion of this collection of atoms and molecules. Otherwise known as the bulk.
So even though we were able to more or less link the atomic and the molecular spectra to more or less equivalent mechanisms (transitions between quantized energy levels (this quantization comes from quantum mechanics)). A further "connection" to a black-body spectrum is not possible (because it is a concept of thermodynamics).
It's where the energy comes from that determines the spectral characteristics. For black-body radiation it is the random collective motion of bulk material constituents. This motion is responsible for an object's temperature which is a statistical measure.
Molecular spectra as well as atomic spectra come from the motions of isolated atoms or molecules that are excited beyond their ground states, but still discrete quantum states. For molecules, it is mostly vibrational or rotational motions of atoms that make up the molecules. For atoms, it is the electrons jumping from one orbital to another.