  This is an extended and corrected version of the author's Diplomarbeit. A
class of algebras called generic pro-$p$ Hecke algebras is introduced,
enlarging the class of generic Hecke algebras by considering certain extensions
of (extended) Coxeter groups. Examples of generic pro-$p$ Hecke algebras are
given by pro-$p$-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion
of an orientation of a Coxeter group is introduced and used to define
`Bernstein maps' intimately related to Bernstein's presentation and to
Cherednik's cocycle. It is shown that certain relations in the Hecke algebra
hold true, equivalent to Bernstein's relations in the case of Iwahori-Hecke
algebras.
  For a certain subclass called affine pro-$p$ Hecke algebras, containing
Iwahori-Hecke and pro-$p$-Iwahori Hecke algebras, an explicit canonical and
integral basis of the center is constructed and finiteness results are proved
about the center and the module-structure of the algebra over its center,
recovering results of Bernstein-Zelevinsky-Lusztig and Vign\'eras.
