Just to expand upon "What are embedding exactly" since the rest seems pretty well answered.
If I mention the word orange out of context, you might associate it with various things. Orange is a colour. Orange is a fruit. Strangely, the fruit also seems to kind of relate to the colour. But also, Orange is a telephone mobile network (or was, anyway).
We could begin to say then, that there's a higher association between the word orange being a colour, which might be 95% of how I'd probably use the word orange, maybe 20% of the time I mean the fruits, and maybe 0.001% of the time I mean the mobile network.
So if broadly I ask the computer, "What invoices have I stored recently for my bills", and I have three notes by way of example, one a review about the film A Clockwork Orange, the other a recipe book, and the third a collection of monthly bills, the computer might be able to determine that the Orange network is a telecom bill and present that note, while avoiding bringing up recipes for fruit salads, because the embeddings are what allows for that kind of associativity on a probability level. When it gets more advanced, words can pair up with other words, so the computer might know "Clockwork Orange" refers to a film more than it's likely to refer to a literal orange constructed out of iron and motors, and that context begins to shift probability in more basic ways than me being vague above saying I mean the colour 95% of the time.
By itself the embeddings don't actually do anything and are just an abstract series of numbers that humans wouldn't find very intuitive, but they're what's needed for computers to begin understanding more complex associations in the case above for semantic search (and others) to function efficiently, by precomputing a lot of this association so that it doesn't have to be done over and over for efficiency. The semantic search then can operate on this embeddings index to be speedy, but you could perhaps have other smart stuff go on, like say automatic tagging of related notes by subjects, people in them, & etc.
So in simple terms, an embeddings index is a specific type of database that allows for the computer to understand deeper concepts behind language rather than just raw text; more mathematically, it's a collection of vectors that associates words to other words in super duper high dimensional space.
