Is light affected by gravity? Why? I would like to know if light is affected by gravity, also, I would like to know what is the correct definition of gravity:
"A force that attracts bodies with mass" or "a force that attracts bodies with energy, such as light"?
Is light massless after all?
 A: 
I would like to know if light is affected by gravity,

Yes, it is. Its motion is affected by gravity, and it also produces its own gravitational field. Its motion is affected by gravity because, in GR, the gravitational field is actually the geometry of spacetime. Analogous to Newton's first law, all small particles follow geodesics if they aren't acted on by other forces (gravity isn't a force in GR). The fact that the path of light bends when it's near a massive object was one of the first observational tests to determine if GR was an accurate theory.
We know light produces a gravitational field because light is composed of electromagnetic fields, and the stress-energy tensor of the EM field is nonzero.

also, I would like to know what is the more correct for the definition of gravity: A force that attracts bodies with mass or force that attracts bodies with energy, such as light.

If you're talking about Newtonian gravity, then the first definition is accurate. The second definition is wrong in both Newtonian gravity and GR. In GR gravity isn't a force at all; it's a consequence of the geometry of spacetime.

Is light massless afterall?

It depends on your definition of mass. The definition nearly everyone (physicists, chemists, engineers) uses is called "rest mass" or sometimes "invariant mass." It is defined by:
$$m=\frac{1}{c^2}\sqrt{E^2-p^2c^2}$$
where $E$ is total energy, $p$ is the magnitude of momentum, and $c$ is the speed of light. For light, this quantity is zero.
Some people still use a definition of mass that most people believe to be outdated and not particularly useful. It's called "relativistic mass," and is simply defined as:
$$m_{rel}=\frac{E}{c^2}$$
Since $E$ is total energy (including kinetic energy), this version of mass is actually frame-dependent and will increase with increasing velocity. This is rather inconvenient, but since the energy of a photon is nonzero the relativistic mass will also be nonzero.
PS: If you ever hear someone say mass increases with velocity, realize that they are talking about relativistic mass. Pop-sci books and TV shows love to start rambling about increasing mass, even though nobody really uses that definition of mass anymore. Rest mass is always the same at any velocity.
A: I'd like to answer the first question "Is light affect by gravity? Why?" because I remembered an adorable thought experiment.
In the spirit of the equivalence principle consider an observer in a closed box. As we all know, the observer inside that box would be unable to tell (neglecting tidal effects) the difference between a uniformly accelerated box (think rockets) or gravity (think the Earth).
Let's say that the observer is in outer space and he's being accelerated with some rockets. Inside the box, he shoots a beam of light from one wall to the other across the rocket, in the direction perpendicular to the direction of perceived inertial acceleration/direction of travel. It's obvious that the light will have a curved trajectory because as it was travelling from one wall to the other, the rocket moved a bit.
By the equivalence principle, the light should also bend if the observer was in a box on the surface of Earth, because otherwise, he would have a way to distinguish a uniformly accelerated frame from a frame in a gravitational field.
What did I tell you, adorable, right? :)
A: Light is affected by gravity, since the source of gravity is mass "curving" spacetime, and light ordinarily travels in straight lines. However, curved spacetime means that straight lines are no longer straight, meaning that a "well" in spacetime formed by a mass causes light to bend around it.
As to whether photons have a mass, that is quite complicated. One could say that the photon does have mass because a photon has energy $E=hf$ where h is Planck's constant and f is the frequency of the photon, and energy is equivalent to mass according to Einstein's famous formula $E=mc^2$. 
However, in modern terminology, the mass of an object is its invariant mass, which is zero for a photon.
 this can be taken to mean different things if the light is moving freely or trapped in a container.  The definition of the invariant mass of an object is $m= \sqrt{ \frac{E^2}{c^4} - \frac{p^2}{c^2} } $.  By this definition, a beam of light is massless like the photons it is composed of.  However, if light is trapped in a box with "ideal" mirrors so the photons are continually reflected back and forth in both directions symmetrically in the box, then the total momentum is zero in the box's frame of reference but the energy is not.  Therefore the light adds a small contribution to the mass of the box.  This could be measured--in principle at least--either by the greater force required to accelerate the box, or by an increase in its gravitational pull.  You might say that the light in the box has mass, but it would be more correct to say that the light contributes to the total mass of the box of light.  You should not use this to justify the statement that light has mass in general.
