As mentioned by
knzhou, a prompt lepton is a lepton originating from the main collision taking place in the event, as a direct product of the particular decay you're after. An analysis looking for a particular final state containing leptons is really looking for prompt leptons.
Non-prompt leptons come later: either through the decay of the hadronized quarks (so-called jets), or a as a "mis-ID". The former basically means that e.g. a $B$-hadron coming from a $b$-quark might decay, before or within the detector, into one or more leptons. These will leave tracks and hits in the relevant parts of the detector as would a prompt lepton, but correlating those hits with nearby jets or $b$-tagged activity could be a way of removing those unwanted non-prompt leptons.
It could also be the case that due to a particular jet signature or a fault in a part of the detector, a jet is reconstructed as a lepton. These "mis-ID" or "fake" leptons are also considered non-prompt.