This is drawing from some recent US history classes and reading about British history, hopefully it’s up to snuff.

The history of the British Empire is long and complicated, and went through many phases. Having violently seceded from the Empire in the American Revolution, the US tried to distance itself from Europe, famously adopting the Munroe Doctrine declaring that the new world was for America and the old for the European powers. However, most other colonies didn’t have nearly such an acrimonious divorce from Britain. Canada was made independent by the British North America act of 1867, and New Zealand followed suit in 1907, both becoming the self governing “dominions” you asked about. The rest of the empire disintegrated diplomatically around the mid 20th century, between the mid ‘40s and ‘60s. Many of these nations joined a diplomatic entity called the Commonwealth of Nations, basically a club of former British colonies sharing the same (politically impotent) monarch adorning currencies. The US, already a major power very distinct from the UK, did not join.

Direct answer then: the US became independent following an armed revolution. The “Dominions” were made independent by act of Parliament, acquiring their name in those acts.