The project itself is stored in a QGIS project file that saves all the necessary links. Copy this to the new computer and open it with QGIS. The double click an a project when you open QGIS is only a list of "recently used" projects that QGIS proposes you to open. It only links to the QGIS project file you saved somewhere.
See also How to Share a QGIS Project
QGIS projects are saved in a dedicate project-file type with extension qgz
(or qgs
). This can be found on your drive wherever you have saved it. If you're not sure about the location, go to Menu Project
-> Save As...
, then you should see where it has probably been saved.
For the project file, see here fore details and about what kind of information is stored there and Working with Project Files from the official QGIS documentation.
The project file itself does not contain any data (be it vector, raster, attributes etc.), but contains a reference to the file, URL or network address where this data can be found. If you move the project file to another place, links to local data files (layers) could be broken. So it is crucial to have a good data management plan.
So make sure to reproduce the exactly same folder and file structure/hierarchy on the new computer, otherwise locally saved data might not be found. Under Menu Settings
-> Options
-> General
-> Default paths
you can define if QGIS uses relative or absolute paths.