Yes. Software is installed locally.
You can subscribe to CreativeCloud (CC) for LR and PS. LR is available as standalone (perpetual license) also.
Here's a guide on how to move over: http://www.lightroomqueen.com/ready-move-aperture-iphoto/
Plenty of information about Lightroom's file structure, and the difference between "Folders" and "Collections". Suggest reading up on the differences and running tests beforehand.