Strictly speaking, fluorescence is emission of a photon from an excited singlet state to a ground singlet state, and phosphorescence is emission of a photon from an excited triplet to a ground singlet. Due to the spin-forbidden nature of the latter transition, phosphorescence is usually slow/delayed, and frequently weak. It may also not be observed due to oxygen quenching, and is almost always very red-shifted compared to the maximum of the low-energy absorption band. For the purposes of your ocular response, fluorescence is instantaneous and disappears as soon as you remove the light source. Ironically, phosphorescence has nothing to do with phosphorus, even though that's what it was named after.
Luminescece is a broader term that includes both fluorescence and phosphorescence as well as other related photon emission mechanisms like chemiluminescence or bioluminescence. It can also be used to refer to some situations where neither fluorescence and phosphorescence formally applies, such as in the luminescence of some metal-ligand charge transfer complexes, which are neither formal singlets or triplets when excited.
Emission can be used generically to refer to any of these, as well as other processes (incandescence). It's a casual, loosely defined term that is probably better avoided in formal publications.
(As for instrumentation, any of these can be measured with a fluorimeter, but to distinguish between them absolutely, you usually need some kind of time-resolved equipment or some other type of experiment that can be used to determine the nature of the emitting state.)