Romans are people from / related to the city of Rome, which is a major city that's been around for thousands of years. It's still important today, as the capital of Italy.

Based on your question, I'm guessing you're thinking of the ancient Romans. They started out just in Rome and the surrounding countryside, but they gradually expanded their control until they ruled over the entire Mediterranean and many regions beyond. This was the Roman Republic for part of its existence and then the Roman Empire (and then other things). Sparta and Egypt were two of the places Rome ruled over, along with many others.

The Roman Republic/Empire started around 500 BC and lasted until about 500 AD, a period of a thousand years.

Egyptians are people from Egypt, which is a country/culture/civilization from the northeastern corner of Africa, centered around the Nile river.

Ancient Egypt is one of the earliest civilizations known to history. It was around thousands of years before Rome was founded. Egypt was ruled over by many different dynasties of kings (or pharaohs) and was conquered several times by outside forces. Eventually one of those outside conquerors was Rome, which ruled Egypt for centuries. After the Romans, Egypt was conquered several more times. It's an independent country today.

Egyptian civilization started around 3100 BC. Egypt was conquered by Rome in 30 BC, over three thousand years later.

Spartans are the people of Sparta, a city in ancient Greece. Sparta was what we call a city-state, a country that consists of basically just a single city. (Singapore is a good example of a modern-day city-state.)

Sparta was one of the major Greek-speaking city states for a long time, warring with other Greek cities such as Athens and Corinth. After many centuries of classical Greek civilization, Greece was conquered by the Romans. Sparta gradually lost importance until it was basically abandoned.

Sparta was founded sometime before 600 BC. It was taken by Rome in 146 BC.