By Don Koks, 2018.
This question is not easy to answer. In our modern time, a century after relativity was formulated, special relativity is usually understood to mean the study of motion in the absence of gravity (that is, in flat spacetime). General relativity is usually understood to mean the study of true gravity (that is, curved spacetime). In particular, the study of accelerated frames in the absence of gravity tends to be classed nowadays as special relativity.
But the distinction seems not always to have been this way. Pais's biography of Einstein states that when Einstein first discussed acceleration in relativity, he wanted to generalise his special theory to non-inertial frames. With this history in mind, it might be said that special relativity should be considered to relate only to inertial frames, and that general relativity contains everything else, from accelerated frames to gravity.
When Einstein started thinking about acceleration, he saw an immediate connection to gravity (the real gravity that makes tidal forces); this prompted him to postulate the equivalence principle. Modern writers have followed suit, by tending to discuss acceleration only superficially, and then segueing quickly into a discussion of gravity proper. The upshot is that in the eyes of some authors, discussions of accelerated frames have become inseparable from discussions of gravity. And yet accelerated frames and gravity are quite distinct from each other. The study of accelerated frames does not require any notion of gravity.
A problem of pedagogy then appears: the language gets confusing. You can find any number of discussions of the Twin Paradox that say "To analyse the situation from the viewpoint of the accelerating twin, general relativity is needed". It's not clear what that sentence really means. If it means that a discussion of accelerated frames is needed, then it's true. If it means that a discussion of true gravity (tidal forces) is needed, then it's false. (We must distinguish the "pseudo gravity" that appears in an accelerating frame—the "g force" that we feel while being accelerated—from the real gravity that is produced by mass and energy. Pseudo gravity has nothing to do with mass or tidal forces.) So, the sentence "General relativity is needed to analyse the Twin Paradox" becomes ill defined, and is best avoided. It's certainly true that most of those who say general relativity is needed in the Twin Paradox don't really know what that sentence means; they are just repeating something they heard elsewhere. It's also true that some who use that phrase mistakenly think that an analysis of gravity is needed to resolve the Twin Paradox, even though they have never seen any such analysis.
Accelerated frames are routinely handled using the theory of inertial frames: see the FAQ "Can Special Relativity Handle Acceleration?". It turns out that the true gravity (tidal forces) that arises from mass and energy invites us to consider spacetime as curved, unlike the case for inertial frames. The result is that both inertial and accelerated frames are handled fully by flat spacetime, whereas true gravity requires curved spacetime. So we now have a neat, well-defined separation of scenarios into flat and curved spacetime, and this is why the meanings of "special" and "general" relativity have evolved to refer to curvature rather than frames. "Special relativity" is nowadays understood to refer to flat spacetime—which can certainly handle accelerated frames with their pseudo gravity. Special relativity is then sufficient to explain the Twin Paradox. (We just need to tell that to all those physics explainers on Youtube who are busy telling their audience that the explanation of the Twin Paradox requires general relativity.) And "General relativity" is nowadays understood to refer to curved spacetime, meaning the study of true gravity that arises from mass and energy.
These modern meanings of "special" and "general" are apparently not what Einstein had in mind, since he couldn't predict any future difficulties in language and interpretation when he first spoke of generalising special relativity; he tended to discuss acceleration in the same breath as true gravity. But from a modern viewpoint, we no longer set "special = inertial" and "general = non-inertial". Instead, we set "special = flat spacetime" and "general = curved spacetime", in the hope that this will minimise any confusion of the role played by acceleration in gravity. Unfortunately, that confusion is still very much present in the subject. It is the reason that we should be very explicit about what we mean when discussing these topics.