Who is Fatah and why are they fighting with Hamas?

Answer 1

Fatah is a secular Palestinian organization,
established in the late '60 ; its aim is to build
a Palestinian country in the occupied territories
(that Israel has 'taken' in 67); they do recognize Israel
as a country and since the Oslo agreement in '92 they
act on a political level, unlike Hamas that acts on a
military level, and does not recognize Israel.

The conflict between them is mainly due to wanting
leadrship / control over the Palestinian people.

Answer 2

They are an opposing Palestinian faction/party to Hamas.