The controversy concerned the significance of infinities in quantum field theory. The problem of infinities was anticipated in the first papers on quantum field theory by Heisenberg and Pauli, and then in 1930 infinite energy shifts were found in calculations of the effects of emitting and reabsorbing photons by free or bound electrons, by Waller and Oppenheimer. In both cases you have to integrate over the momenta of the photons, and the integrals diverge. During the 1930s it was widely believed that these infinities signified a breakdown of quantum electrodynamics at energies of the order of 100 MeV. This changed after the war, when new techniques of calculation were developed that manifestly preserved the principles of special relativity at every step, and it was recognized that the infinities could be absorbed into a redefinition, called a renormalization, of physical constants like the charge and mass of the electron. Dyson was able to show (with some technicalities cleared up later by Salam and me) that in quantum electrodynamics and a limited class of other theories, the renormalization of a finite number of physical parameters would actually remove infinities in every order of perturbation theory — that is, in every term when we write any physical observable as an expansion in powers of the charge of the electron, or powers of similar parameters in other theories. Theories in which infinities are removed in this way are known as renormalizable. They can be recognized by the property that in renormalizable theories, in natural units in which Planck’s constant and the speed of light are unity, all of the constants multiplying terms in the Lagrangian are just pure numbers, like the charge of the electron, or have the units of positive powers of energy, like particle masses, but not negative powers of energy.
Read the whole essay if you have time. He made it quite "easy" to comprehend.