Thaumaturgy, in 3e, is being born with the ability to learn random little magical tricks, as well as the knowledge of one such trick. Examples from the corebook include reading tea leaves, knowing how to arrange a firepit just so to light a fire without flint or tinder that rain or similar won't put out, the ability to conduct an exorcism or the ability to break a piece of bread into two new loaves (very subtle refernce there, that last one).
Honestly, I like the idea as a standalone thing - some people just have these weird things they can do. That's interesting. However, what I don't like is the fact that it replaces systematic study of the magical weirdness of Creation, which was a really cool thing. My personal response to the thing has been to keep the 3e Thaumaturgy as a thing, and have most of the stuff that 2e Thaumaturgy could do (with a few exceptions, such as weather-working, which you do by asking the gods very nicely) be simple applications of Craft (for talismans, potions and such) and Occult, albeit requiring specific sources (e.g. demonomicons, books of alchemy etc) and often exotic ingredients. For example, anyone can simply know that ghosts can't cross lines of salt or germinated grain, and thus can lay a 'ward against ghosts'. Whether they'll do it right or not is another matter entirely, and the more involved procedures do require substantial background knowledge (i.e. 3+ dots in Occult/Craft/Lore/whatever), but it's possible.
Plus, this system opens up the possibility of a true savant deciphering parts or the whole of how a thaumaturge's 'tricks' work, allowing them to systemize it properly, though the opaqueness of any given mini-miracle and the difficulty/stuff required for a non-thaumaturge to use it would depend on Storyteller fiat. Second Bread, for instance, would probably have to have some esoteric components for a non-thaumaturge to do it, be incredibly rare or else be really, really hard to figure out, because economics. It'd probably be more common for a savant to work out bits and pieces, and fit those bits and pieces together with others to create entirely new, only vaguely related procedures.