Alchemy is a bit hard to define in a single sentence due to the fact that different groups at different times considered alchemy to be about different things. In the simplest sense, alchemy was the art of trying to attain a level of perfection in both objects and humans. For objects, this most famously includes turning base metals into precious metals like gold. For humans, this includes finding and ingesting a fabled "elixir of life" which would grant long lasting health and vitality to the drinker. Many alchemists attributed a deep spiritual connection to their work. Alchemy was the start of chemical research, and some alchemists (like Robert Boyle) became known as the fathers of modern chemistry.
In terms of turning base metals into gold, the only chemical research done today that would come close to this is nuclear chemistry or particle physics. Turning one element into another requires the adding or subtracting of protons from the nucleus of an element. Standard chemical reactions deal with electrons, not protons, since electrons are much easier to manipulate. The only real way to alter the protons in a nucleus is through either nuclear fusion or fission whereby one element is converted to another. Accelerating particles in a particle collider to smash them together and make new elements is also in this same vein.
In terms of the elixir of life, traditional and herbal medicine probably come the closest combining mystical health with spirituality. But this obviously falls way outside the boundaries of modern chemistry.