I presume the answer is that it depends on the mass and size of the star and black hole and how they approach either other, but I was wondering if somebody could provide some rough bounds (e.g. hours vs thousands of years) theorized or based on historical observed data.
By "eating" I mean the time it takes a star to go through the event horizon of the black hole and potentially reach the singularity.
One of the main ways black holes are noticed is by looking at a solar system where the star appears to move as though it were a binary star system (e.i. two stars) when only one is seen. In these situations, depending on the distances, the black hole "feeds" off the original star, and a stream of the stellar plasma is slowly pealed off the star into the black hole.
This matter can sometimes form a very vivid accretion disk, that can be observed using telescopes (see Herbig–Haro object). This process can take a very long time, on the order of millions of years. However, of course, a rogue black hole could enter a star system head on and collide right with the sun and "suck it up," which would happen rather quickly (to an outside observer).
If you are asking about a star being "eaten" by a supermassive black hole at the center of a galaxy, as is being in another question here on Physics SE due to a recent event covered in the popular press (GRB 110328A), this relatively authoritative paper says that they stay bright for one month to one year.
This long period is due to the orbiting of the torn up parts of the star.
Probing Quiescent Massive Black Holes:
Insights from Tidal Disruption Events
A Whitepaper Submitted to the Decadal Survey Committee
