An inauguration is a ceremony that formally marks the start of something. You’ve probably seen the inauguration ceremony when a new president takes office.
Inauguration can refer to a formal ceremony, such as one marking the start of a new term in office or the opening of a new wing in a museum, but it can be used more broadly to describe a less formal beginning. Your boss may make you responsible for the inauguration of a new system at work. There probably won’t be a ceremony for that inauguration — in this case, inauguration just indicates that a new practice is being introduced.