A latent variable is an unobserved factor, like a personality trait or attitude, that is inferred statistically from patterns among variables that can be directly observed.
"Latent" simply means hidden or not directly measurable. You can't ask someone to report their Extraversion score the way you'd ask their height, so instead you ask a set of related, observable questions — do you enjoy parties, do you speak up in groups, do you seek excitement — and use the statistical pattern among the answers to estimate the hidden variable behind them. The individual answers are called manifest, or observed, variables; the trait they're all pointing to is the latent one.
This is the same basic logic behind factor analysis: a latent variable is essentially what a "factor" is once you've identified it and given it a name, like Extraversion or Conscientiousness. The math doesn't know or care what to call the pattern it finds — that interpretive step is a human judgment, made by looking at which observed items cluster together most strongly.
Nearly every construct in personality psychology is a latent variable in this technical sense, whether or not anyone uses the term day to day.