Your question seems to be very simple on the surface, but it is in fact rather complicated. Before I answer your question, I would like to tell you a short story about the famous physicist Richard Feynman. I am not sure that I remember all the details correctly, but they are not all that important to the main point of the story. One day when he was young, he was walking in the woods with his father and young Richard saw a bird and he asked his father,"What's the name of that bird?" His dad studied birds, and he told Richard that the bird was a sparrow. He also told Richard the different names for the bird in several different languages: Japanese, Spanish,
Russian, Latin ... . Then he said, "Well now, I have given you many different names for the bird, but remember that you really don't know anything more about the bird except for its name. The name of the bird is not important. What is important is how the bird flys, how it builds its nest, how it migrates, how it lays its eggs, ... ."

So, what does this story have to do with the mass of light? Well if you read a textbook, you will find out that light can be thought of as being made up of particles called photons. These photons have no mass! So there is the simple answer, light has no mass and no weight. This is a lot like knowing the name of the bird, you know that light has no mass and no weight, but you don't know how light behaves.

Well it turns out that even though light has no mass, it is attracted by gravity. This effect is so small that we are not aware of it in ordinary light, but astronomers must keep this in mind, because there are certain situations when this effect will be observable. Now in order to see this effect, you need a large body with a lot of mass which will bend the path of light due to its gravity. One example of a big body is the Sun, but since the Sun gives off so much light itself, it is often difficult to see this. It turns out that during a total eclipse of the Sun, scientists can see the effect of the Sun's gravity bending starlight from distant stars. Incidentally, this effect was first predicted by Einstein, but scientists had to wait several months for the next eclipse to see if he was right. (He was, of course, correct.)

I hope you understand a little more about light, and also a little bit about how science sometimes has the "simple" answer that is good enough to explain things most of the time, but that sometimes things get to be a little more complicated in very special situations.

Light is believed to be massless.According to what is known as the Particle Data Book, a summary of what has been measured in particle physics, the upper limit to the mass of a photon (a light "particle") is extremely small. Unfortunately, I don't know how the measurement was done. If I can find out, I'll let you know.

Light does not have mass, and therefore there is nothing for the force of gravity to pull on, so light doesn't have weight either.
However, light does have energy. Think of the oscillations that cause water waves at the beach. The oscillations don't have mass, but they do have energy, which you've surely felt if you've ever been hit by a big wave.
Light can be thought of as either an oscillation or as a particle, called a photon. In order to measure the energy of a beam of light, you could put a detector in a box with a small hole in one side, and shine the light beam through the hole onto the detector.. The energy that the detector sees will be the energy of a photon of the light times the rate of photons passing through the hole in the box times the time that the light beam is turned on. The energy of a photon corresponds to the wavelength of the light oscillations, or the color of the light. Which do you think has more energy, a photon or red light or a photon of blue light? The rate of photons passing through the hole in the box corresponds to the intensity of the light beam. How is the power written on a lightbulb (such as 60W or 100W) related to the intensity of the light it produces?
What sorts of materials would make good detectors for light? Would a mirror work? How could you use a black piece of paper to detect light?

Light is very special. It has properties of both waves and particles, for example, it will bend around corners to a small degree due to diffraction as a wave. On the other hand, if you lower the intensity of a source enough (make it dim enough) you can actually count the arrival of photons which compose the light. These photons behave like particles. Here is the tricky answer to your question-- light doesn't have mass (at least it must be very, very small) but it does have signifcant momentum. So, for example, a beam of light reflecting off a mirror pushes on the mirror, but a very small amount. One way to measure the momentum is with a mirror, placed on a very sensitive balance in a vacuum jar to avoid air currents. Then a strong beam of light will push on the mirror, and one can find the momentum by relating the pressure with the intensity. This is the idea behind a "solar sail" -- a very thin mirror, thousands of meters wide which can be used to move space-craft using the light pressure of the sun. On the other hand, light does have energy -- and Einstein showed that energy and mass were related-- so in that sense, light does have mass -- but it still doesn't have "Rest Mass". i.e. a photon always travels at the speed of light -- if you try to slow one down, it merely loses energy (but doesn't go slower) until you have nothing left. Light traveling up from the earth into space, actually shifts to lower energies due to the effect of earth's gravity -- but the effect is very small. You can't measure it for visible light, but you can for gamma rays which are much higher energy light particles.
Sunlight, generated near the core of the sun, can take 100,000 years to escape the surface. The pressure deep in the sun is so high, that the light hits matter and is scattered in random directions. It gets scattered so much that although it is still going the speed of light, thousands of years elapse before it gets to the surface and can esacpe. This means that the sun has over 50,000 years of generated light trapped within. This light creates pressure which makes the sun larger than it would be otherwise. In fact, for larger stars, the light pressure is larger than any other source of pressure in the star. For super big stars (200x the mass of the sun) the light pressure defeats the gravity holding the star together -- and the star falls apart.

So, light has no rest mass, but it does have momentum and energy-- it can be measured by sensitive balances or by the diameter of stars!

Most people believe that light has no mass but it is still a good idea to check.

It turns out that the way the upper limit on the photon (a light "particle") mass was determined was by making measurements of the Earth's (and I've heard Jupiter's) magnetic field. If photons have mass then magnetic fields at large distances will behave differently than if photons don't have mass. The experiements did not see any of the effects that photons with mass would have, so the scientists involved were able to put a limit on how much mass a photon could have based on how good the instrument was. It is an extremely low limit, so if photons did have mass, they would be very light.

Another way to try to measure if photons have mass is to try to measure the speed of light for different types of light (for example radio waves, visible light, and x-rays). If the speed of light is different for different types of light then photons do have mass. How hard do you think it would be to measure the difference in the speed light between different types of light? I hesitated to mention this possibility earlier, but I heard from a professor here at UCSB that people have actually tried to do this experiment. He didn't know what the results were though.