There are many proofs.

(First proof)

• Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Akad. Leipzig 88 (1936), 141–164 (German)

(Thurston's rediscovery)

• William P. Thurston and John W. Milnor, The Geometry and Topology of Three-Manifolds

(Variational principle)

• Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669 (French).
• Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994), 553–580.
• Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689.
• (see also) Günter M. Ziegler, Convex polytopes: extremal constructions and f-vector shapes, Geometric Combinatorics, 2007, pp. 617–691.

(An inductive proof ?)

• Kenneth Stephenson, Introduction to Circle Packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005.

(I also recommend the following completion of the theorem)

• Graham R. Brightwell and Edward R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (1993), no. 2, 214–229.

There are many proofs, and I'm not claiming that the following list is complete. New references are welcome.

(First proof)

• Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Akad. Leipzig 88 (1936), 141–164 (German)

(Thurston's rediscovery and related)

• Andreev, E. M., Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (1970), no. 125, 256–260.
• (see also) Roeder, Roland K.W., Hubbard, John H. and Dunbar, William D., Andreev’s Theorem on hyperbolic polyhedra, Annales de l’institut Fourier 57 (2007), no. 3, 825–882.
• William P. Thurston and John W. Milnor, The Geometry and Topology of Three-Manifolds

(Variational principle)

• Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669 (French).
• Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994), 553–580.
• Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689.
• (see also) Günter M. Ziegler, Convex polytopes: extremal constructions and f-vector shapes, Geometric Combinatorics, 2007, pp. 617–691.

(An inductive proof ?)

• Kenneth Stephenson, Introduction to Circle Packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005.

(I also recommend the following completion of the theorem)

• Graham R. Brightwell and Edward R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (1993), no. 2, 214–229.

There are many proofs.

(First proof)

• Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Akad. Leipzig 88 (1936), 141–164 (German)

(Thurston's rediscovery)

• William P. Thurston and John W. Milnor, The Geometry and Topology of Three-Manifolds

(Variational principle)

• Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669 (French).
• Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689.
• (see also) Günter M. Ziegler, Convex polytopes: extremal constructions and f-vector shapes, Geometric Combinatorics, 2007, pp. 617–691.

(An inductive proof ?)

• Kenneth Stephenson, Introduction to Circle Packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005.

(I also recommend the following completion of the theorem)

• Graham R. Brightwell and Edward R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (1993), no. 2, 214–229.

There are many proofs.

(First proof)

• Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Akad. Leipzig 88 (1936), 141–164 (German)

(Thurston's rediscovery)

• William P. Thurston and John W. Milnor, The Geometry and Topology of Three-Manifolds

(Variational principle)

• Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669 (French).
• Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994), 553–580.
• Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689.
• (see also) Günter M. Ziegler, Convex polytopes: extremal constructions and f-vector shapes, Geometric Combinatorics, 2007, pp. 617–691.

(An inductive proof ?)

• Kenneth Stephenson, Introduction to Circle Packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005.

(I also recommend the following completion of the theorem)

• Graham R. Brightwell and Edward R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (1993), no. 2, 214–229.