What is the thing on a common laptop called that replaces a mouse?

A touchpad (also trackpad) is a pointing device consisting of specialized surface that can translate the motion and position of a user's fingers to a relative position on screen. They are a common feature of laptop computers and also used as a substitute for a computer mouse where desk space is scarce.

https://en.wikipedia.org/wiki/Touchpad

While I would agree that touchpad seem to be becoming more prevalent, there's still quite a few out there using what is called a pointing stick.

From https://en.wikipedia.org/wiki/Pointing_stick:

"The pointing stick (trademarked by IBM as the TrackPoint & by Synaptics as the Touchstyk) is an isometric joystick used as a pointing device (compare especially touchpad and trackball). It was invented by research scientist Ted Selker. It is present on many brands of laptops, including IBM's line of ThinkPad laptops (now made by Lenovo), Toshiba Tecra laptops, HP business notebooks and on Dell Latitudes under the name of Track Stick."

My latest laptop has no touchpad (thank goodness, I hated it and disabled it instantly on getting a laptop), just the TrackPoint device (commonly referred to as a joystick), which is fantastic - right in the middle of the keyboard, so you can use a finger of either hand to control it, a thumb of either the same or the opposite hand to hit the 'button', and don't have to lift your hand out of typing position to do so! Note the red button in the keyboard on this picture

https://www.laptopmag.com/reviews/laptops/lenovo-thinkpad-x61s