The term "to turn over a new leaf" is used to refer to making a new start. It is often used specifically to describe changes in personal behavior that are made with the goal of being a better person. Many people are encouraged to do this if they struggle in interpersonal relationships, as a way of asking them to think about how their own behaviors might be influencing their interactions with other people.
From the sound of it, one might think that this phrase is related to freshly budding green leaves in the spring, which carry a pleasant image of renewal and fresh starts. The term actually refers to turning the page of a book, however, and it dates to the 1500s. The implication is that one is turning over the previous page with the bad behavior, and starting anew on a fresh page.
Allegedly, when accused of recidivism after promising to change, Oscar Wilde once said that he earnestly intended to turn over a new leaf, but he hadn't gotten to the bottom of the page yet. He supposedly assured his critic that as soon as he was done, he would make good on his promise.
In the context of personal behavior, turning over a new leaf can take a number of forms. Many New Year's resolutions are a form of new beginning, with people vowing to make changes in their lifestyles which are intended to be beneficial. People might also vow at any time of the year to make changes in the way they treat coworkers or employees with the goal of making their working environments more pleasant, or to change their attitude to authority figures in the hopes of getting along better.
The term is also sometimes used more generally to refer to entities, rather than specific individuals. For example, a city that radically changed its policies on some issue might be said to be turning a new leaf, and the term would also be used to reference improvement.
Many people believe that any time is a good time to turn over a new leaf, and that there is always room for personal improvement. By carefully examining one's past relationships and present behavior, sometimes it is possible to find an area for improvement that could be addressed by resolving to change and start a fresh page.