Particles can be caracterized, in particular, by their mass m and their decay lifetime Δt. The decay lifetime is the average time interval taken by the particle to decay to lighter particles. The lifetime of a particle can be infinite (e.g., for the electron) if there is no lighter particles to decay to, but can also be very small (e.g., for the Higgs boson or the Z boson) in case many decay channels are open. If the lifetime Δt of a particle is small, the Heisenberg principle tells us that its mass m is uncertain:
Δt x Δm ~ h/2π
The uncertainty on the mass is called the decay width (or simply width) of the particle, and is denoted Γ (pronounce "Gamma"). For example, the Z boson has a mass of ~91.2 GeV and a width of ~2.5 GeV. It means that the Z boson decays so fast that it can be produced with a mass of 91.2 ± 2.5 GeV. Similarly, the Higgs boson has a mass of 125.5 GeV and a width of 4.2 MeV (0.0042 GeV). It decays about 500 times less quickly than a Z (although still extremely fast), hence its mass is much better defined.
Now, the Z boson can decay into a pair of neutrinos. It actually happens in about 20% of the Z decays. By definition, the partial width of the Z boson into neutrinos is 20% of the total decay width ΓZ. It turns out that the neutrinos are mostly "invisible" to us, in the sense that they interact extremely weakly with matter: they cannot be detected in the detectors built for high-energy colliders. By extension, the partial width of the Z boson into neutrino pairs (and any other pair of invisible particles, should they exist and be light enough) is called the invisible width of the Z boson.
The Higgs boson can also decay "invisibly" through the decay channel H -> ZZ, with both Z's decaying into a neutrino pair, but this invisible width is expected to be very small in the standard model (less than 0.1% of the total width ΓH.) Should the invisible width of the Higgs boson be larger, it would be an essential discovery of a new physics phenomenon.
NB. Although the decay products are invisible, it turns out that the invisible width of a particle can be measured via different methods. For example, the Higgs boson is produced at e+e- colliders in association with a Z boson: e+e- -> ZH. If the Higgs boson decays invisibly, such a collision will be identified with the presence of a Z boson (decaying for example in a pair of charged leptons), and a missing mass of 125.5 GeV. The probability that such a configuration occurs directly gives the fraction of the times with which the Higgs boson decays invisibly, hence its invisible width.