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Genocide as a term came into being relatively after World War 2. There used to be a term earlier called Crimes against Humanity which used to be used to describe similar events. This use of this term has decreased over a period of time and the term Genocide is used increasingly in modern political parlance.
Genocide is a Crime against Humanity.
If you look at the root of the word genocide, the Latin word genus, you see that genocide is about a group of people, in whole or in part, but in either case targeting them because of their group membership. Ethnic group, 'race,' nationality, religion.
Crimes against Humanity which would not be genocide are those where a systematic policy is in place, yet that doesn't target ethnic or similar groups. An attempt to kill all concert pianists would not be genocide because pianists are no genus, but it might be a Crime against Humanity if the attacks are no isolated incidents.
The international definition of the crime of genocide is supplied by the 1948 Convention on the Prevention and Punishment of the Crime of Genocide. It proscribes a collection of acts
...with intent to destroy, in whole or in part, a national, ethnical, racial or religious group...
The "in whole or in part" clause gives some breadth to what can be considered genocide, legally.
Since your question is about the political use of the term, the legal definition rapidly becomes meaningless since political speech has never been truly bound by legal definitions of any kind and instead obeys the rules of rhetoric. Genocide, therefore, gets tossed about roughly as loosely as 'terrorist,' insofar as it is important to note the distinction between someone's rhetorical bloviation and an actual legal claim when understanding how each term is being used. (See This question for more examples of this.)
"Crime against humanity" doesn't have a clear pedigree in the same way, but is inclusive of genocide, terrorism, piracy, human trafficking, and other large-scale insults to the dignity of the common human condition. It's lack of specific codification makes it ripe for rhetorical convenience.
The definition of fundamental human rights co-evolved alongside these terms, and isn't directly informative to them per se, but would definitely be relied upon by any prosecutor trying to make such a case in court, particularly Article 3:
Everyone has the right to life, liberty and security of person.
and when necessary, Article 4:
No one shall be held in slavery or servitude; slavery and the slave trade shall be prohibited in all their forms.
Somewhat simpler/realistic example: (under ICC definitions) a campaign of detentions and torture against (say) socialists (or more broadly regime opponents) would often not rise to the bar of genocide, but would easily qualify as crimes against humanity. Case in point, a German court held that against a Syrian colonel accused of being involved is such a mass torture program under the Assad regime. He was tried under German law, but the German law definitions (section 7 of the VStGB) are close enough to those used by the ICC--art 7 of the Rome statute. Both specifically mention widespread use of torture, for instance.
