The phrase "do your level best" is another way of saying to do your very best. This phrase is a figurative expression that is commonly used in some regions instead of similar literal phrases that have the same meaning. For example, if a person is taking an important test, someone who cares about him may tell him to do his level best. Likewise, if an individual is doing a favor for another but feels unsure of his ability to succeed, the recipient of the favor may ask him to do his level best. Since this phrase is used in figuratively rather than literally, it is referred to as an idiom.
In many cases, people incorporate idioms into their speech and casual writing. Idioms are creative ways of expressing oneself and allow a person to share a thought, feeling, or desire, without having to use literal phrasing. Some people use idioms because they find this type of speech fun while others use them out of habit because they are so common where they live. The phrase "do your level best" is one such idiom. It means to do one's very best or put one's utmost effort into an action or task.
An individual might use the phrase "do your level best" when he is trying to encourage another party to perform well in some capacity. For example, an individual might feel anxiety over his ability to pass a test or give a speech. In such a case, a person who cares about him may want to encourage him to do well by telling him to do his level best. This implies that no matter what the outcome, the effort a person puts into an action is worth it and valuable as long as he tries his hardest.
Sometimes a person will also use the phrase "do your level best" when he is asking another party to help him in some way. For example, if a person is having trouble with a mechanical device, he may ask a party with good technical skills to fix it for him. If that party is unsure of what has caused the problem with the device or his ability to fix it, he may express doubt about the outcome of his efforts. In such a case, the person in need of help may ask him to do his level best. He would be using this phrase to indicate that he understands that the device may prove impossible to fix but hopes the person he's asked for help will put maximum effort into it.