Then it's simple.
You'll, need to move:
1) The actual image files. Keeping them organized in the same folder structure. Where they are to be found on the old computer, depends on what OS, and how you set them up to begin with.
2) The Lightroom catalog database file(s). Again the location depends on the OS, and how you originally set it up. You may have just one or many. These are identified by an *.lrcat filetype.
3) The Lightroom preferences and presets. These are always in the same location, but that depends on what OS.
4) Reinstall the Lightroom application on the new machine. You're allowed individual non-simultaneous use on any combination of two: Mac/PC/Laptop/Desktop.
All of your current editing work will be preseved via #2.
It's really pretty simple, the key is preserving the folder structure of the original photo files.
Tell us more about OS, etc, and we can give you the default file locations.