MissConductUS explained the term "downtown" for big cities, but I'll give an example from the modest urban area in which I grew up. There was a city of about 150,000 people, an adjacent smaller city of about 50,000, and several small towns, most of which had started out as farming communities but were evolving into suburbs. At the center of the larger city was the state capitol building, surrounded by grassy lawns. On a few adjacent streets, on each side, were office buildings associated with the state government, several large banks, a large hotel, and a few streets full of restaurants, bakeries, and fun little shops. One of these streets was pedestrianized. The community college was also nearby. This was "downtown," because it was the hub of the city, and it was walkable. Other parts of the city, comprising residential neighborhoods, shopping malls, factories, and the airport, were designed mainly for cars.
The adjacent smaller city centered around a large university. Along one side of campus ran a long street lined with restaurants, cafes, bookstores, drugstores, bars, etc. The business area stretched a block or two deep, and we also called this "downtown" because it was a convenient, walkable shopping area. Beyond that was a large residential neighborhood, and then a high-speed road that was also lined with businesses, but was designed only for driving, not walking.
The origin of the term "downtown" is murky. It may come from Boston, where the city center is situated in a basin, surrounded by hills. Or it may come from New York, where the city center lies south of the big residential neighborhoods and so is "down" on a map. The Boston origin is probably more likely.