On Linux or BSD, the ptrace system call allows one process to take control of another process; it can then observe and control the other process. Breakpoints are inserted by replacing an instruction with a special instruction that traps; the debugger can then take control. Watchpoints (the article calls them data breakpoints) are often implemented by making the containing page read-only, so that a write access traps (this means that there's overhead from other writes to the same page, the debugger has to just resume execution silently for those). A checkpoint can be implemented by forking the process and freezing the fork, so execution can go back to that point.