The wastegate controls how much exhaust gas goes through the turbo to spin it. It keeps the turbo spinning at the proper speed.
A BOV goes between the turbo and the engine's intake. When your turbo is spinning and building boost, and you let off the gas, the throttle body blades slam shut. All the pressurized air has nowhere to go, but back out through the turbo. The BOV has a vacuum line hooked to the engine intake. When there's pressure at the BOV, but vacuum in the engine (like when the throttle slams shut), the BOV opens up and vents that extra pressure out to the atmosphere instead of forcing it back out through the turbo.
A BOV only does anything when there's a pressure difference between the intake (before the engine) and the engine (the intake manifold). And basically the only time that happens is when you let off the gas. It helps more with a manual transmission, where you're on and off the gas a lot to shift. Letting the extra pressure out also means the turbo doesn't get slowed down as much, so it doesn't have to spool back up when you get on the gas. But yeah, a BOV only does anything when you let off the gas.
A popoff valve is similar to a BOV, except that it just opens up at __psi. It's just a spring valve that prevents you from getting over __psi. They still control boost, but you need a wastegate to control how fast the turbo spins, so that it's not out of its efficiency range and making even more heat.
