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 in all dimensions, and there are finitely many examples in the complex hyperbolic case (at least for $\mathbb{C}$-dim >1) going back to Deligne and Mostow (see also [Thurston's paper][5]). However, Gromov and Schoen have shown that quaternionic and Cayley-hyperbolic lattices are all arithmetic, so the Borel Harish-Chandra construction is complete in these cases.  

Addendum: Related to Protsak's comment, there is a simple example of a negatively curved homogeneous space which has no lattice action. This is Thurston's "9th geometry", which is excluded as a geometry precisely for this reason. One can take the double warped product metric $$ dr^2 + e^{2a r} dx^2 + e^{2b r} dy^2, $$ for $a,b >0$. When $a=b$, this gives hyperbolic space. But when $a\neq b$, the sectional curvatures are $-a^2, -b^2, -ab$. This has a solvable transitive group of isometries, so is homogeneous. But using the solvability, one may see that there is no cocompact action. (Remark:  when $a=-b$, this gives sol geometry).


  [1]: http://www.ams.org/mathscinet-getitem?mr=147566
  [2]: http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html
  [3]: https://en.wikipedia.org/wiki/Mahler's_compactness_theorem
  [4]: http://www.ams.org/mathscinet-getitem?mr=932135
  [5]: https://msp.org/gtm/1998/01/p025.xhtml "William P. Thurston: Shapes of polyhedra and triangulations of the sphere"