Traditionally, *nix kernels (including Linux) spawn the init process with PID 1 during system boot, which in turn starts other user-mode processes and is considered the root of the process tree (all the other processes start below this process in the tree). The PID namespace allows a process to spin off a new tree of processes under it with its own root process (PID 1 process). PID namespaces isolate process ID numbers, and allow duplication of PID numbers across different PID namespaces, which means that processes in different PID namespaces can have the same process ID. The process IDs within a PID namespace are unique, and are assigned sequentially starting with PID 1.
PID namespaces are used in containers (lightweight virtualization solution) to migrate a container with a process tree, onto a different host system without any changes to PIDs.