Motivated forgetting is what Freud referred to as repressing memories. According to Freud, there are times when an event or an action is so painful that we can't deal with the memory of it, so we repress the memory completely. By pushing the memory into the subconscious and actively repressing it, we are unable to recall the memory. So in essence, motivated forgetting is purposeful forgetting, even if it's not purposeful at a conscious level. Here's an example at a conscious level. Tonight, I want you to do whatever you have to do in order to forget the following numbers: 5-3-1. Do whatever you have to, but forget the numbers "5-3-1" by this time tomorrow. If you can do it, you will have experienced one type of motivated forgetting.
Note: if you actively try to forget these numbers, you probably won't be able to. So while some memories might be so painful they are repressed in an unconscious process, the idea of conscious, motivated forgetting is pretty questionable.