In particle decays, what is the relation between partial widths and branching ratios?

If you don't know what the term width refers to, please read first this answer.

As mentioned there, the decay width Γ of particle is directly related to its decay lifetime: the faster the particle decays, the larger its decay width. In general, a particle can decay in several "modes", also called "decay channels". For example, a Z boson can decay into a pair of neutrinos, a pair of charged leptons, or a pair of quarks (i.e., all the standard model fermions lighter than mZ/2). The probability for a Z to decay into a neutrino pair is about 20%, into a pair of charged leptons (electrons, muons, or taus) is about 10%, and into a pair of quarks (u,d,c,s,b) is about 70%. These probabilities are also called branching fractions or branching ratios.

The partial width of a given decay channel is nothing but the product of the total width Γ and the corresponding branching ratios. For example, the partial width of the Z into neutrinos (also called invisible width of the Z) is 20% x 2.5 GeV, i.e. about 500 MeV. The sum of the partial widths of a particle equals the total width Γ.