An informal term meaning first right of refusal within a given context. The right of first and exclusive access to some entity, opportunity or reward which is attainable by two or more individuals but is informally claimed or granted to one as a right - usually based on some pretext. The implication is that if the one with 'first dibbs' declines the opportunity, that subsequent individuals with second or third dibbs then inherit that right in turn.
Johnny Howard: I've got first dibbs on that meat pie because I haven't eaten for seven months.
Kimbo Beastly: You're an idiot and a bullshit artist.
Johnny Howard: That may be very well my dear fat friend but I take it dry from the big George W so I can have first dibbs on any meat pie I want... ahem, if it pleases the house.