On Tarski's “The Semantic Conception of Truth and the Foundations of Semantics,” (1944), Hartry Field's “Tarski's Theory of Truth” (1972), and Donald Davidson's“The Folly of Trying to Define Truth” (1977).
What is truth? Tarski gives a technical, metaphysically neutral definition for truth within a particular, well-defined language. In short, a sentence is true if it is "satisfied" by all terms in the language. A sentence with a variable would be satisfied only by certain terms: "x is red" is satisfied only by the red ones. But a sentence without a variable, if true, is satisfied by all terms: "Rose (a) is red," where (a) is a particular rose, is satisfied by that rose, because it's red, but it's also technically satisfied by my dog, by you, by Trump, etc., because none of those things interferes with the sentence being true. He had some lengthy proofs and logical language to establish this, and yes, this is weird.
One of his main reasons for doing this was to rule out things like the liar's paradox: e.g., "This sentence is false." On Tarski's analysis, a well-defined language doesn't allow this kind of sentence. As with Russell's analysis of the paradoxical "sets that aren't members of themselves," Tarski distinguishes a language from its meta-language: When you say "'Snow is white' is true," you're taking a sentence in the language (snow is white) and by putting it in quotes, you're making it an object and saying something about it, using a different language, the meta-language, which includes all the terms of the object language, but also semantic terms like "is true." Tarski's truth definition is only about an object language (which cannot include any self-reflexive terms of this sort), but is stated in a meta-language. To make this clearer, some explanations actually use different languages, e.g., "'Schnee ist weiss' is true if and only if snow is white." German here is the object language, while English is the meta-language. (But of course, as I said, everyday German couldn't really be an object language, as it's not well-formed in the required way. There would have to be no room for misunderstanding.)
Tarski claimed to be capturing our everyday notion of truth and making it more precise, with the idea that this could be extended from the formal, logical languages he was dealing with to at least formalized vocabularies within the sciences.
Field (treated along with Davidson in the second part of our discussion) charges Tarski with misunderstanding his own project. He says that Tarski was, like many of his peers in the era of behaviorism, wary of semantic notions, i.e., he wanted to reduce talk of mind to talk of matter.
To explain: The definition I've given above applies to just one single sentence. The definition of truth for the whole language is just the collection of true sentences so defined. This is an "extensional" definition, kind of like defining "sheep" not by giving essential characteristics of a sheep, but by pointing to the whole collection of sheep. Of course, if a new animal comes along, this leaves you unable to tell if it's a sheep or not. So it's not a definition as you'd normally consider it. And Tarski's definition only then is talking about actual sentences in a particular language, and isn't giving a general definition for truth for all languages. This would require, e.g., the semantic notion of synonymy (meaning), which (on Field's analysis) is an off-limits for Tarski.
Anyway, does just pointing to a bunch of (ultimately physical) objects and saying "those are the true ones" perform the reduction that Field says Tarski is trying to accomplish? No, because (Field says) there's still the semantic notion of reference smuggled into Tarski's procedure. We can better interpret what Tarski is doing by saying that he's defining truth in terms of other semantic notions, i.e., reference, definition, and satisfaction, which Field still thinks is helpful.