Well, energy can come in countless different forms. As heat, as light, as a current through a wire, as mass, as a speeding car, ...
What's special is that mass is just another form of energy. You have probably heard of E = mc^2, but what does it mean? It means that anything with mass actually contains an enormous amount of energy, and the conversion between the two is the square of velocity of light. In fact, it's possible to use a system of units in which distance and time are measured in the same units, so that c = 1. In that system, E = m. They're the same.
Conversely, energy can be converted into mass. In the interior of the Sun, hydrogen nuclei (protons) are being pushed together to form helium nuclei. The resulting helium is lighter than the hydrogen with which you started, the rest of the mass is converted into heat and light.
Going a little more mundane, because E=mc^2, it's also true that m = E/c^2. So a hot potato will weigh more than a cold potato, because it holds more energy. Let's assume you put a cold potato in the microwave oven for a minute at 1000 Watts. That'll radiate about 60 kiloJoules of energy into the potato. Using m = E/c^2 this gives us a mass difference of m = 0.667590034 nanograms. Such a small weight difference would be practically impossible to measure on something as big as a potato.
So to answer your question: Yes, it is possible, and with carefully set up experiments it's possible to check the equivalence between mass and energy. But in more mundane situations, the weight differences are so small that there is no practical way of measuring them, besides inferring them from energy differences.