Underway means in motion or in progress. When a ship is underway, it has left the dock and is moving through the water. When a project is underway, work has started and things are happening.

Today we use it more broadly. A school play becomes underway when the curtain rises and the first scene begins. A construction project gets underway when workers arrive and start digging or building.

You'll often hear phrases like “the game is underway” (it has started) or “plans are underway” (they're actively being carried out and moving forward, beyond the discussion stage). The word implies that something is actively moving forward, that momentum has built. When your teacher announces that testing is underway, it means the tests have started and are happening right now, with students actively taking them.

Notice that underway describes the middle phase of something: past the planning stage, past the starting gun, but not yet finished. It's the active, rolling-forward part of any endeavor.