Shortly after German Botanist Matthias Schleiden discovered that plants are made of cells, Schwann discovered that animals are as well (Source).
He hypothesized that fermentation wasn't a reaction with nitrogen in the air as people then believed, but that tiny organisms (yeast) were eating the sugars of the fruit and then excreting carbon dioxide and ethanol. People in his day noticed yeast multiplied with fermentation, but he was the first to suggest they were the thing doing the fermenting, rather than just a byproduct of it. He also stated that yeast were living plant-like organisms, firmly establishing fermentation as a biological process. He was publicly ridiculed for this hypothesis which came at the end of his short career and probably helped lead to its demise, as he could no longer get funds or promotions to continue doing scientific research. (Paraphrased from The Other Brain, by R. Douglas Fields, 2009, and Wikipedia- Theodore Schwann.)
He suspected that hydrochloric acid wasn't the whole story with digestion. He discovered Pepsin, the enzyme that helps break down proteins (The Other Brain). He also showed that bile is essential to digestion, and coined the term "metabolism" describing chemical changes in animal tissue (Source).
The myelinating cells in the Peripheral Nervous System are his namesake, since he first described them. He studied nerves, noticing they had little dew-drop like structures running the length of them. He thought that a neuron being built during development must join with other cells to create the long axon by fusion, and the Schwann cells were remnants of this process from those fetal cells. Although his speculation was incorrect, these cells still carry his name.