I will have some of each listed below and you can make the choice as which one you would like to try out on your PC. There are also some Linux distributions that are full blown Linux distributions yet still are lightweight as they run lightweight Linux software, you can still install any of the non-lightweight Linux programs as well. Some lightweight Linux distributions are not full blown Linux distributions and only have a set number of Linux programs that can be installed easily without having to install the software manually. You can even install them on your fast computers to make them even faster! Remember that almost all Linux distributions can be made lightweight by installing lightweight Linux programs instead of running bloated Linux programs. There are many distributions that can be installed on on your slower computers to make them run faster.
After adding the information about lightweight Linux apps, I figured I should write about some of the lightweight Linux distros.