How does ArcMap do what? It can display data without a spatial reference / coordinate system (SR / CS), but try to add other data that's really in a different CS. They're not going to overlay unless you happen to set the data frame's coordinate system to the unknown data's CS. ArcMap does not reproject data that has no spatial reference / coordinate system. ArcMap simply displays it.
When you define the coordinate system, you're just updating the metadata. In essence, you're saying, "I know/think/hope that this is the correct coordinate system for the data."
It's often pretty easy to figure out the coordinate system for a dataset if it's projected, although you can still get the geographic coordinate system portion wrong. Get the projected coordinate system wrong, the data can end up thousands of meters or feet offset.
If the data's in latitude/longitude, decimal degrees, it's easier to make a mistake because at most the data may be offset a few hundred meters, and is often much closer. Without excellent reference data to compare against, you're just guessing.
Note: The best solution is to track down the data provider and get them to \c\o\u\g\h \u\p tell you what the coordinate system is.