What is the role of conjectures in modern mathematics?

From wikipedia:

Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that (amongst other things) makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true. In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called conditional proofs: the conjectures assumed appear in the hypotheses of the theorem, for the time being.

These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type. Instead, a conjecture is considered proven only when it has been shown that it is logically impossible for it to be false.

But on the other hand, not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generally accepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner (much as we can take Euclid's parallel postulate as either true or false).

The word "conjecture" is rather fuzzy and doesn't in itself tell you much. It can be used about just about every statement where

1. Someone whose judgment you respect thinks it is likely to be true,
2. No proof of it is known, but
3. It feels like the kind of statement that ought to be subject to proof if it is true.

Thus, simply being told that "such-and-such is a conjecture" doesn't tell you much useful.

Conjectures play at least two different roles in mathematical research:

• They're goals we set ourselves to have something to strive for. Often these are fairly simple statements that give the mathematician the impression that they ought to have a proof or disproof, but where we simply don't have the tools to attack them. So we set out trying to invent such tools!

  Goldbach's conjecture falls into this category, as does, for example, the twin prime conjecture or (until it was proved) Fermat's Last Theorem. These are things that really won't have any particularly important consequences, but it is hoped that searching for techniques that can whack them will also be actually useful for less famous but more practical purposes.

  Sometimes these get resolved by proving that they cannot be proved from a reasonable set of assumptions (so condition 3 above is not satisfied). This famously happened to the continuum hypothesis, almost a century after it was first conjectured, when Paul Cohen showed that it doesn't follow from the usual axioms of set theory.

• They're stepping stones towards what we really want to know. This is a matter of division of labor: A community of researchers want to investigate this-or-that, and a respected and experienced person suggests that it ought to be possible to prove such-and-such and then prove that such-and-such implies this-or-that. If the suggestion is accepted, people can now work independently on proving such-and-such and on proving the step from such-and-such to this-or-that, and the Such-and-Such conjecture is now the point that connects these two efforts.

  This can sometimes result in the Such-and-Such Conjecture being famous for its own sake, particularly if the step from such-and-such to this-or-that gets completed, but proving such-and-such itself turns out to be hard. (That is, without uncovering evidence that such-and-such is simply false).

Note that the terminology here is not very consistent. Even though it is now common to speak of this general kind of claims as "conjectures", particular named conjectures need not have "conjecture" in their name. Some are named Hypothesis instead (and this doesn't encode any particular different meaning, but is just a historical accident), and Fermat's Last Theorem spuriously had "theorem" in its name for several centuries before it was actually proved.

A conjecture is an unproved theorem. Since it's unproved, you can't use a conjecture to prove a theorem or solve a problem. But some conjectures are famous enough to get their own names so that they can be referred to easily.

More importantly, conjectures often represent an area of research that a community of mathematicians will work on. Many differential geometers worked on the Poincaré conjecture until it was proven by Grigori Perelman. Andrew Wiles and Richard Taylor's proof of Fermat's Last Theorem was actually a proof of the Taniyama-Shimura-Weil conjecture. The Langlands program is a set of conjectures that has directed number theory for decades. So conjectures serve as goals for mathematicians to work towards.

It can happen that conjectures are proven false as well, which is significant too.

Once upon a time there was a conjecture known as The inner function conjecture for several complex variables.

Now, an inner function $f$ for the unit disc $\mathbb{D}=\{z\in\mathbb{C}: \,|z|<1\}$ is an analytic function such that $|f(z)|<1$ for all $z\in \mathbb{D}$, that is $f:\mathbb{D}\to\mathbb{D}$ and such that the radial limit towards the boundary is $1$ almost everywhere, i.e. $$\lim_{r\nearrow1}|f(re^{i\theta})|=1,\qquad\textrm{for almost all $\theta$}.$$ These functions are very useful complex analysis and there are fine factorization results on e.g. the Hardy space. For example, the zeros can be factored into a Blaschke product (which is an inner function).

The inner function conjecture asserts there are no non-constant inner functions on $\mathbb{B}^n$, the unit ball of $\mathbb{C}^n$ ($n>1$). That is, if $f:\mathbb{B}^n\to\mathbb{D}$ is analytic then it is constant. (I think it was conjectured by W. Rudin in 1966).

Unable to prove the theorem, people found that if there were inner functions then they would behave very strangely.

In 1981 Alexandrov found a way to construct inner functions, thus disproving the conjecture.

Lesson learned: Complex analysis of several variables can be hard.