What is the United Kingdom's connection to the British Empire and the Commonwealth? In which order were the countries of the UK conquered in the nineteenth century?

There are a number of separate but related concepts here that involve the British Isles.
The United Kingdom is the official name of the government that controls England, Scotland, Wales, and Northern Ireland. These four separate nations came together between the 1500s and the 1700s; Ireland was previously part of the United Kingdom from 1801 to 1922, but today it is a sovereign nation. The United Kingdom can be thought of as somewhat similar to the United States: they are a national government binding together different geographic entities, although there are many differences between the two governments' structures.
The United Kingdom also ruled over the larger British Empire, which expanded quickly from the British Isles themselves to include much of North America, India, Australia, regions throughout Africa, and many different islands all over the globe. The British Empire was the largest empire in human history and, at its largest point in 1900, controlled the destiny of one out of every four humans living on the globe.
The Commonwealth is an organization meant to bring together the various nations of the former British Empire who have been granted independence, allowing them to work together for peaceful purposes like trade, cultural exchange, and good relations. Fifty-three nations belong to the Commonwealth (the United States is not one, since we fought the British for independence). Many feature the British sovereign, Queen Elizabeth II, on their money as a testament to their shared history.
The countries of the United Kingdom were not conquered in the nineteenth century, although Ireland was formally introduced to the UK in 1801, having been effectively conquered by the English many years prior. However, in the nineteenth century the UK did conquer many countries, such as India and South Africa, and these became part of the British Empire.