How exactly does the Spring BeanPostProcessor work?

The typical example for a bean post processor is when you want to wrap the original bean in a proxy instance, e.g. when using the @Transactional annotation.

The bean post processor will be handed the original instance of the bean, it may call any methods on the target, but it also gets to return the actual bean instance that should be bound in the application context, which means that it can actually return any object it wants. The typical scenario when this is useful is when the bean post processor wraps the target in a proxy instance. All invocations on the bean bound in application context will pass through the proxy, and the proxy then gets to perform some magic before and/or after invocations on the target bean, e.g. AOP or transaction management.

Spring doc explains the BPPs under Customizing beans using BeanPostProcessor. BPP beans are a special kind of beans that get created before any other beans and interact with newly created beans. With this construct, Spring gives you means to hook-up to and customize the lifecycle behavior simply by implementing a BeanPostProcessor yourself.

Having a custom BPP like

public class CustomBeanPostProcessor implements BeanPostProcessor {

    public CustomBeanPostProcessor() {
        System.out.println("0. Spring calls constructor");
    }

    @Override
    public Object postProcessBeforeInitialization(Object bean, String beanName)
            throws BeansException {
        System.out.println(bean.getClass() + "  " + beanName);
        return bean;
    }

    @Override
    public Object postProcessAfterInitialization(Object bean, String beanName)
            throws BeansException {
        System.out.println(bean.getClass() + "  " + beanName);
        return bean;
    }
}

would be called and print out the class and bean name for every created bean.

To undersand how the method fit the bean's lifecycle, and when exactly the method's get called check the docs

postProcessBeforeInitialization(Object bean, String beanName) Apply this BeanPostProcessor to the given new bean instance before any bean initialization callbacks (like InitializingBean's afterPropertiesSet or a custom init-method).

postProcessAfterInitialization(Object bean, String beanName) Apply this BeanPostProcessor to the given new bean instance after any bean initialization callbacks (like InitializingBean's afterPropertiesSet or a custom init-method).

The important bit is also that

The bean will already be populated with property values.

For what concerns the relation with the @PostConstruct note that this annotation is a convenient way of declaring a postProcessAfterInitialization method, and Spring becomes aware of it when you either by registerCommonAnnotationBeanPostProcessor or specify the <context:annotation-config /> in bean configuration file. Whether the @PostConstruct method will execute before or after any other postProcessAfterInitialization depends on the order property

You can configure multiple BeanPostProcessor instances, and you can control the order in which these BeanPostProcessors execute by setting the order property.