By default, yes, LR saves the editing and other metadata in the catalog file. To "share" it to the other computer you can either copy the catalog file/folder to the second computer or use an external hard drive to keep your catalog and image files and just unplug it from the one and plug it into the other.
You can locate your catalog file/folder by going to Edit/Catalog Settings, and in the General tab/dialog it will show the location. You can copy using your system browser, but care should be taken to keep the images and catalog locations consistent, one reason why keeping the browser and image files on the one drive.
The alternative approach is to use the Lightroom "Save Metadata to File(s)" function. This function will write "sidecar" files alongside of all your selected images which contain the editing and other metadata info. So, to get that data to the other computer, copy the image folders over then in the other computer use the LR "Read Metadata from File". Or, if you are working on new images, you can use the LR Import function in the second computer and it should "pick up" the metadata.