The examples are arithmetic groups, constructed in general by [Borel and Harish-Chandra][1]. 
See also [Dave Witte Morris' preliminary book][2]. 
However, examples in hyperbolic and complex hyperbolic spaces probably go back further to the study of quadratic forms. 

For hyperbolic lattices, one can take a quadratic form over a quadratic number field (such as $\mathbb{Q}(\sqrt{2})$), which
is Lorentzian at one place, and definite at the other places (such as $x_1^2+\cdots +x_n^2-\sqrt{2} x_{n+1}^2$), and take the group of matrices $\Gamma$ in $GL(n+1,\mathbb{Z}[\sqrt{2}])$ which preserve this quadratic form. Then [Mahler's compactness theorem][3] (cf. Witte Morris) implies 
that the quotient $\mathbb{H}^n/\Gamma$ is compact. Then by Selberg's Lemma and residual finiteness (as Greg points out, Malcev's Theorem), you may find a torsion-free subgroup 
of finite-index with as large injectivity radius as you like. 

For hyperbolic and complex-hyperbolic spaces, there are other examples which don't come from the arithmetic construction (in fact, most hyperbolic surfaces and 3-manifolds are not arithmetic). These are attributable to [Gromov and Piatetski-Shapiro][4] in the hyperbolic case, and there are finitely many examples in the complex hyperbolic case (at least for $\mathbb{C}$-dim >1) going back to Picard. However, Corlette has shown that quaternionic and Cayley-hyperbolic lattices are all arithmetic, so the Borel Harish-Chandra construction is complete.  


  [1]: http://www.ams.org/mathscinet-getitem?mr=147566
  [2]: http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html
  [3]: http://en.wikipedia.org/wiki/Mahler's_compactness_theorem
  [4]: http://www.ams.org/mathscinet-getitem?mr=932135