When to capitalize "Republic":
* As part of a proper noun: You would capitalize "Republic" when it's part of a specific country's name:
* The Republic of Ireland
* The Dominican Republic
* The People's Republic of China
* At the beginning of a sentence: This is a general rule of grammar, but it applies to any word.
* Republic is a form of government.
When not to capitalize "Republic":
* As a general term: When referring to the concept of a republic in general, you don't capitalize it.
* The country is a republic, with a president as its head of state.
* As part of a common phrase: These phrases use "republic" in a general way.
* The Fifth Republic of France (this refers to a period in French history, not a specific country)
* The Roman Republic (refers to a historical period, not a specific country)
In short: Capitalize "Republic" when it's part of a proper noun (like a country's name) and at the start of a sentence. Otherwise, leave it lowercase.