Let's try to break down this definition into small pieces.
UniqueEvery project is different. This is a rule. Even If you do something more than one time , you should know that you won't face the same difficulties and problems.
For Example : Say you have an office that offer designs for houses. Each construction project is just a unique project. Weather can change your design. Also may be you should change the concrete used depending on where it is built or you may need to change the roof structure to handle snow.
Specific GoalEach project should has a specific goal to accomplish. May be the project is needed to solve a problem in your company like reducing the costs or marketing your products in different way to increase sales.
Beginning and EndIf the project doesn't has an end , this is because the goal isn't clear enough. The goal of project is crucial to identify when it is needed to be done.
BudgetMost of the time we are thinking of budget as money but in fact it can be the resources needed for your projects and time.
Note : You should note that there is difference between a project and normal operations
Example : If you are a student , your default work is to study every day , attend lectures , do assignments and reports but if the professor asked you to do a research about a new topic. This can be called a project in that case.