Can someone provide real example please?

Level Of Effort is used to estimate project hours. As I've used it, it is a more precise estimate than a High Level Estimate. The LOE is done after the project is under way and you know more about it. Typically the estimating process is: Rough Order of Magnitude for the initial discussions, HLE when the stakeholders are considering funding and LOE after the project starts.

The only apportioning I've done is where I had to share a resource across projects. You estimate the work that resource has to do and work with the other PM to make sure they are not over allocated and can complete both project's tasks.

Discrete efforts is where you isolate the work required to deliver one thing - function, feature, or whatever and then determine the estimate for that work. Discrete efforts are typically used to understand the impact of small tasks.

Examples:
Discrete effort - 100 hours will be required for qualification test XYZ
Apportioned effort - The test plan author requires 100 hours to develop 10 discrete test plans at approximately 10 hours each including test XYZ and others.
Level of Effort - 2 lab technicians are assigned full time for 3 months to support all test related activities.