There are a few ways tea and coffee are decaffeinated. But first, the interesting facts.
1. Decaffeination does not take place efficiently without some water of solvation. That is, while caffeine is soluble in solvents such as methylene chloride, the leaf/bean must be just a slight bit wet to fully extract. This also means that the final product usually needs to be dried afterward. All decaffeination must overcome a number of complicating factors.
2. no solvent is perfect. Other things, such as the aroma of the tea, are soluble in the popular decaffeination solvents, which are supercritical carbon dioxide, ethyl acetate, or (rarely nowadays) some chlorocarbon such as methylene chloride. methylene chloride has the lowest flavor impact largely because fatty aromatic compounds and cuticle waxes are not very soluble in this solvent, and because it has a very low boiling point, and this helps to preserve the flavor. Methylene chloride is also thought to be carcinogenc, though, and is a greenhouse gas contributor, so it is highly unpopular. You would think that carbon dioxide would be perfect, because it is a gas and is widely present in nature. But ironically, this solvent requires the most water to have its effect and is very damaging to tea leaves, and because the end product must be dried much of the aroma is lost as the water is boiled off. Ethyl acetate is another popular solvent because it can be derived from natural sources (expensively) and because caffeine is very strongly soluble in ethyl acetate. But for tea, this is a bad choice because the polyphenols of tea (thought to give it its healthful properties) are also soluble in ethyl acetate and there is a noticeable impact upon the color and flavor.
3. adding water means the tea or coffee must be dried. Unfortunately, the wetting and drying often on a microscopic scale causes damage and partial extraction, leaving many times a fine coating of 'instant' tea or coffee on the surface. But because the dried solids from partial extraction have not been treated, the redrying process often leaves them in a state which makes them much harder to re-extract in hot water.
4. no solution is perfect. In an ideal world, we can understand solution quite easily. Dissolve sugar in water, and some more, and some more, and at some point you cannot dissolve any more. But that's easy to understand, because there's only sugar and water, and only one phase change (solid to solution). But with caffeine extraction, the problem is considerably more complex. In aqueous solution, caffeine will begin to self-associate while still in solution, so as caffeine is more concentrated, it is actually harder to extract despite the fact that it is more soluble (if a solid instead of a solution) in the extracting liquid. Add in a solid solution phase (the leaf or bean) and all the other components possibly that can dissolve, and what you are looking at is not a perfect solution.
5. solutions are much easier to decaffeinate. Because a solution alone removes some of the complexity, it is far easier to decaffeinate instant tea or coffee. Caffeine can be removed by adsorbing it on solid filters which trap caffeine, such as XAD4 or activated carbon. But, once you've made instant tea, you've lost the idea of the end consumer "fresh brewing" their decaf coffee. Most instant tea and coffee is decaffeinated with solid resins while in solution because it is the cheapest process to run.
6. some people think decaf tea or coffee is weak. The irony is that because decaffeination is expensive, tea companies use less decaf leaf in a bag of tea and that might be why some decaf tea is perceived as weak. Because of the impact of decaffeination, special blends are used and there is some definite loss of flavor and solids, but all in all, if done well decaf tea and coffee where all other variables are controlled have very small differences from their caffeinated origins. It is often a problem of blend- cheaper beans and leaves are used to start, and cheap tea and coffee tastes, well, cheap.
The vast majority of decaf is done with supercritical carbon dioxide. It's ostensibly natural, non-hazardous, but the cost of installing a CO2 plant is very expensive.
There is an alternative, known as the "swiss water" process. It combines the technology of resin based extraction with the fact that water is an excellent solvent for caffeine. Tea or coffee is brewed, and the brewed tea/coffee is decaffeinated over a resin like charcoal. The decaf tea/coffee solution is then passed over fresh leaf. In theory, there is "room" for the caffeine but the tea or coffee is fully saturated so that it cannot dissolve off the leaf or bean. In practice, this may be an acceptable, non chemical method but the result is far from perfect, mostly because water is very difficult to remove and the water that is present is so soaked with dissolved solids that in essence, the end result may be no different than drying instant tea/coffee on used leaves/grounds. I also take exception to this on the basis that decaffeinating resins are far from natural and often contain carcinogens themselves that can be leached out into the end process. But because of technicalities, I think, swiss water process is not held to the same standard as solvent extraction because the carcinogens do not "directly" touch the leaf or bean. No, just the same water is washed over both.
Ethyl acetate is still used, but due to the overwhelming consumer demand for "natural" tea or coffee, even natural source ethyl acetate isn't much competition.
Personally, I avoid all form of decaffeination. If caffeine is bothering you, drink less tea or coffee. Hey, look! I've managed to shake the party line!