The Boltzmann distribution gives the probability of a system to be in a given state (specified by the coordinates and momenta of its constituents) as proportional to the exponential function of the energy of that state, in units of kT.

For calculating the exact probability for a given range of states, you must normalize the distribution, i.e., you must divide by the total integral for all states (be careful, not the same as "for all energies").

Partition function is the name for this total integral. It is given an special name because, in fact, mathematically, you can derive a lot of information from this integral. For example: to calculate the mean value of a given function of the state, the standar formula of integrating the product of the function by the probability, and dividing by the partition function, is the same as the derivative of the logarithm of the partition function.