An affiliation is an official connection to something. To have an affiliation to something is to be affiliated with it. The Tea Party pretty clearly has an affiliation with the Republican Party.

You can be connected to all sorts of things, but to have an affiliation is to have an official connection. A local church might have an affiliation with a larger religious organization. A bunch of small groups might share an affiliation with a larger one, as in the case of baseball teams and the leagues they belong to. Often you'll hear politicians denying any affiliation with certain groups, places, or companies, because they want to appear neutral. If you have an affiliation with something, you're likely to pay it extra attention.

Definitions of affiliation
  1. noun
    the act of becoming formally connected or joined
    “welcomed the affiliation of the research center with the university”
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    affiliation anew
    type of:
    the act of consorting with or joining with others
  2. noun
    a social or business relationship
    “a valuable financial affiliation
    synonyms: association, tie, tie-up
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    type of:
    a state involving mutual dealings between people or parties or countries
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