What exactly is a "garbled circuit"? There are plenty of questions here about the details and how-to's of "garbled circuits", but I have not seen anything that defines what garbled circuits are.

What exactly is a garbled circuit? What are they intended to be used for? What are their limitations? 

The tag for garbled circuits just says that they are used in secure multiparty computation. However, this answer states that they're only appropriate for two-party computation?

This question seeks the definition of a "circuit". What is the difference between a circuit and a garbled circuit?
A circuit is just a way to represent a computation. There is nothing specifically cryptographic about a circuit. It just means a straight-line computation (no looping or flow-control constructs) consisting of just operations on bits, like AND, OR, NOT.

A garbled circuit is a way to "encrypt a computation" that reveals only the output of the computation, but reveals nothing about the inputs or any intermediate values. We use the term "circuit" because garbled circuits work by taking the computation you care about, expressed as a circuit, and then doing some cryptographic stuff for each operation (AND, OR, NOT) in the circuit.

If we want to be a little more precise, a "garbling scheme" consists of:


(Garble) A way to convert a (plain) circuit $C$ into a garbled circuit $\widehat C$.
(Encode) A way to convert any (plain) input $x$ for the circuit into a garbled input $\widehat x$. You need the secret randomness that was used to garble the circuit to encode $x$ into $\widehat x$.
(Evaluate) A way to take a garbled circuit $\widehat C$ and garbled input $\widehat x$ and compute the circuit output $C(x)$. Anyone can do this, you don't have to know $x$ or the secret randomness inside $\widehat C$ to evaluate and learn $C(x)$.


I'm simplifying a little bit here. But the main idea of security is that $\widehat C$ and $\widehat x$ together leak no more information than $C(x)$. In particular, they reveal nothing about $x$, yet they allow the computation $C(x)$ to be done (obliviously). This is what I mean by "encrypting a computation".

The main application for garbled circuits is secure two-party computation. Imagine that Alice has private input $x$ and Bob has private input $y$. They agree on some function $f$ and agree that they both want to learn $f(x,y)$, but don't want their opponent to learn anything more than $f(x,y)$. To achieve this, they can do the following (this is Yao's classical protocol):


The parties agree on a way to express $f$ as a (plain) circuit. Alice garbles the circuit $f \mapsto \widehat f$. She sends $\widehat f$ to Bob as well as her own "garbled input" $\widehat x$.
Alice knows how to encode any input for $f$ into a "garbled" input, but only Bob knows his private input $y$. So the parties arrange for Bob to pick up a garbled version $\widehat y$ without Alice learning what $y$ was. This can be done with a primitive called oblivious transfer.
Now Bob has the garbled circuit $\widehat f$, and a garbled input $\widehat x, \widehat y$ for that circuit. He can then run the evaluation procedure and learn $f(x,y)$. He can reveal $f(x,y)$ to Alice.


We can argue that the protocol reveals no more than $f(x,y)$ in the following way:


Alice doesn't see anything other than the final answer $f(x,y)$ in this protocol (the security of oblivious transfer ensures that she learns nothing in step 2).
Even though Bob sees $\widehat f$, $\widehat x$ and $\widehat y$, the security of garbled circuits ensures that these values don't reveal anything beyond $f(x,y)$.


This approach works when Alice &amp; Bob are semi-honest (i.e., they follow the protocol as instructed). But when Alice is malicious she can garble some other function $f'$ instead of the $f$ that they agreed upon. So other things need to be added to the protocol to prevent this from happening, when we want security against malicious adversaries.

Reference material:


The Yao Construction and its Proof Of Security
 (video), Yehuda Lindell
A Proof of Yao's Protocol for Secure Two-Party Computation, Yehuda Lindell &amp; Benny Pinkas
Foundations of Garbled Circuits, Mihir Bellare, Viet Tung Hoang, Phillip Rogaway
A Brief History of Practical Garbled Circuit Optimizations, a video of mine. The first several slides cover the "classical" construction of garbled circuits.