Yes, light has 'mass of some sort'. It's not rest mass, but it is a source of gravity. In fact, if you have a box with light bouncing around every which-way inside, the light does contribute to the rest mass of the box.
Anyway, there are two reasons that light bends as it goes around a star, or a black hole, or anything massive. One part you can think of more or less as being due to the mass of the light. (That's mass, not rest mass.) The light accelerates toward the star, the same as anything else. The result is a small velocity at right angles to the initial velocity, so the light has turned toward the star. The second part of the effect comes from the General Relativistic curvature of space. (The first part is also General relativistic, but we're telling a non-relativistic story about it here.) There's really more distance to traverse for the part of the light wave near the star than for the part away from the star! That sounds weird, but it's confirmed. This extra effect causes just as much light bending as the 'normal' effect, so the light bending is twice as big as you might guess.
Both effects are there for any particle, but for ones with rest mass the space-curvature part is much smaller than the 'normal' part, unless they happen to be moving at almost the speed of light relative to the star.
A fully relativistic way of thinking about this is that gravity isn't an "attraction one mass exudes on another", but rather is a distortion of the geometry of space and time. Objects in free-fall do not have forces on them, but follow the local geometry of spacetime, and photons are no exception.
The distortion is determined by the distribution and motion of energy and momentum throughout space. Photons have energy but do not have (rest) mass, and they travel at the speed of light in straight lines. The curious thing is what a straight line is in curved spacetime near a massive object does not correspond to the lines on a grid which you can set up far away from the massive object. In fact, because the paths that photons take in gravitational fields are straight in some coordinates but not in others, the more common name for them is "geodesic curves" or just "geodesics".
Near the sun, the "weak lensing" approximations work, but near a black hole, the full-blown general relativistic treatment is needed. Light can be deflected a small amount if it passes far away from the black hole, bent around a large amount if it gets closer, just plain fall in if it hits head on, or spiral inwards if it strkes the event horizon a glancing blow. You get all of these predictions from Einstein's postulates that the speed of light is the same in all reference frames, and the gravitational equivalence principle that gravity and acceleration are indistinguishable.
(republished on 07/23/06)