People have always been on the move. But if someone leaves their country and settles somewhere else permanently, they are called emigrants (from the country they are leaving) and immigrants (to the country they are going to).
Emigration may be because people want to try to get a better life and more money. This is what happens with most people in poorer countries of the world. People come from North Africa and Asia to Europe and from Mexico and South America to the USA for this reason.
Emigration may be because some natural disaster, like drought, forces them from the place they would otherwise want to stay. Many people left the lands of Africa just south of the Sahara Desert because of this.
Emigration may also be because people are forced away because of war. In recent years very large numbers have left countries like Syria and Somalia to get away from the war zones.
Finally, emigration may be because one country forces some of its people out because it does not want them. Idi Amin of Uganda did this to Ugandan Asians in the 19670s, forcing them to become immigrants to Britain. Jews were forced from Russia at the start of the 19th century and went to the USA, Britain and elsewhere.
There are many examples of this kind up and down the ages.