* Proper Noun: When referring to the state or the city (e.g., "I live in Washington." or "I visited Washington, D.C.")
* Adjective: When describing something related to the state or city (e.g., "Washington politics" or "Washington apples")
* Verb: In some specific cases, it can be a verb, like "to Washington" (e.g., "The troops Washingtoned to the front lines.") This is very uncommon and more of a humorous usage.
To determine the part of speech, you need to look at the sentence and how the word is used.