Space-time bends due to gravity, if that's what you're referring to. This doesn't really lend itself to the idea that light lacks mass, however. The issue of what we're talking about is that quantum physics is inherently illogical and difficult to comprehend. The brain doesn't like to accept wave particle duality because we can't fathom something being in two states simultaneously without actually being either (Schrodinger's Cat). The traditional concept of mass is locked into being viewed as a particle, you can't view a wave as an object with mass regardless of how you look at it. So, regardless of how you look at it, light does not have mass in the same way your keyboard does.
However, it can be inferred that photons carry mass. There's a very, very famous equation that states
E=mc
2. In classical electromagnetic theory (no wave-particle duality)
E can actually be defined as
E=pc where p is merely momentum. However, this equation is limited to the scope of classical electromagnetic theory. Special relativity rejects this equation in favor of
E=mc
2. This was, as you may have guessed, one of Einstein's biggest discoveries.
But it must be such an insignificant amount that it doesn't actually affect anything, right?
Its effect is relative to its mass, as redundant as that sounds. If you drop a baseball on your foot it's going to do significantly less damage than if you drop a bowling ball. Photons are extremely small particles, so the change in velocity an object inherits from being hit by a photon is the definition of negligible. However, photons do exert force on objects they hit.
