Mordell, Diophantine Equations, page 271, writes, "...for the special case $$y^2=2x^4-1$$ it has been known for two centuries that solutions are given by $(x,y)=(1,1)$ and $(13,239)$. It was proved by Ljunggren that these are the only positive integer solutions. The proof is exceedingly complicated." Mordell gives the citation, Zur Theorie der Gleichung $x^2+1=Dy^4$, Avh. Norske Vid. Akad. Oslo No. 5 1 (1942).
Guy, Unsolved Problems In Number Theory, 3rd edition, Problem D6 is "An elementary solution of $x^2=2y^4-1$." Guy cites Steiner and Tzanakis, Simplifying the solution of Ljunggren's equation $X^2+1=2Y^4$, J Number Theory 37 (1991) 123-132, and writes, "Whether Steiner & Tzanakis have simplified the solution may be a matter of taste; they use the theory of linear forms in logarithms of algebraic numbers."
Guy also cites a proof by Chen Jian-Hua, which he calls "unconventional." The bibliographic details are, A new solution of the Diophantine equation $X^2+1=2Y^4$, J Number Theory 48 (1994) 62-74 and A note on the Diophantine equation $x^2+1=dy^4$, Abh. Math. Sem. Univ. Hamburg 64 (1994) 1-10.
I'd suggest also looking at Wikipedia on Ljunggren.
And there's more: Konstantinos A. Draziotis, The Ljunggren equation revisited, Colloq. Math. 109 (2007), no. 1, 9–11.
Michael A. Bennett, Irrationality via the hypergeometric method, Diophantine analysis and related fields—DARF 2007/2008, 7–18, AIP Conf. Proc., 976, Amer. Inst. Phys., Melville, NY, 2008.