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To supplement Geoff's helpful answer, I'd add a further standard reference: Chapter 11 of the 1976 book Character Theory of Finite Groups by I.M. Isaacs (reprinted in an AMS-Chelsea edition). Here and in the old Curtis-Reiner book (Sections 51-53) you get a lot of specifics about how projective representations arise in finite group theory, along with a treatment of the Schur multiplier (and proof that it is also finite). The theory basically originates with Schur a century ago, whose treatment of symmetric groups and related groups is the subject of a 1992 Oxford monograph by P.N. Hoffman and J.F. Humphreys. In modern usage, Schur's "factor sets" get translated into the language of cohomology.

In the case of finite groups, projective representations arise naturally when you have a group and a quotient group to consider, as Geoff points out. But similar ideas occur naturally in physics, for example in dealing with special orthogonal groups as symmetry groups: here representations come up in a physical context and the notion of "spin" of a particle is best explained by lifting a projective representation to the Spin covering group. Now the groups involved may be infinite, but much of the formalism remains the same.

Similarly, when Steinberg set out in his influential 1963 Nagoya paper "Representations of algebraic groups" to study modular representations of Chevalley groups over finite fields, he began by locating projective representations and then investigated how these might lift to covering groups. Here the point is that the interesting groups may be simple, in which case they often have non-simple covering groups in the background: for example, $SL_2(\mathbb{F}_q)$SL_2(\mathbb{F}_q) \rightarrow PSL_2(\mathbb{F}_q)$. Eventually Steinberg's study of liftings and central extensions led to much deeper ideas in the case of infinite fields.

In all of this it's important to avoid any confusion with the unrelated homological language of projective modules, even though people often study finite group representations using module language.
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To supplement Geoff's helpful answer, I'd add a further standard reference: Chapter 11 of the 1976 book Character Theory of Finite Groups by I.M. Isaacs (reprinted in an AMS-Chelsea edition). Here and in the old Curtis-Reiner book (Sections 51-53) you get a lot of specifics about how projective representations arise in finite group theory, along with a treatment of the Schur multiplier (and proof that it is also finite). The theory basically originates with Schur a century ago, whose treatment of symmetric groups and related groups is the subject of a 1992 Oxford monograph by P.N. Hoffman and J.F. Humphreys. In modern usage, Schur's "factor sets" get translated into the language of cohomology.

In the case of finite groups, projective representations arise naturally when you have a group and a quotient group to consider, as Geoff points out. But similar ideas occur naturally in physics, for example in dealing with special orthogonal groups as symmetry groups: here representations come up in a physical context and the notion of "spin" of a particle is best explained by lifting a projective representation to the Spin covering group. Now the groups involved may be infinite, but much of the formalism remains the same.

Similarly, when Steinberg set out in his influential 1963 Nagoya paper to study modular representations of Chevalley groups over finite fields, he began by locating projective representations and then investigated how these might lift to covering groups. Here the point is that the interesting groups may be simple, in which case they often have non-simple covering groups in the background: for example, $SL_2(\mathbb{F}_q)$. Eventually Steinberg's study of liftings and central extensions led to much deeper ideas in the case of infinite fields.

In all of this it's important to avoid any confusion with the unrelated homological language of projective modules, even though people often study finite group representations using module language.