Initially Compendium will have no projects set up.
So the first time you open Compendium you will be presented with the 'New Project' dialog asking you to enter the details to create your first Compendium project:

The 'Project Name' is the name as seen by you, in other words, what do you want to call this project.
If you want this new project to be your default project, then check the box. This means Compendium will automatically launch this project when you open the application. If you check the default project box, it will automatically check the default user box below it. This is because Compendium needs to login to the default database for you to automatically. Note that you can check the default user box independently for any project. This will mean that in the project login window your login details will be automatically filled in for you. (This does not apply to remote MySQL projects for security reasons).
The 'Author Name' is important to complete as it is what you wish to be known as when you export maps and share them with other people. It is this name that is stored against all nodes and links that you create.
The 'Login Name' and 'Password' fields will be used as the login information, which you will need to know to access this project later. This user is set as an 'Administrator' by default.
Once you have completed all the fields, press the 'Create' button and a new project will be created and some default data loaded in.
Once the project has been create you will be presented with a login window, similar to the one below, (though initially only showing your one new project):

For details on the coloured balls and other elements shown in figure 2, see Opening a Project, and for project related issues in general see Project Management.
If you did not check the box to make the user of your new project the default user, you now need to enter the Login Name, and Password as you entered them in figure 1. The login is case-sensitive. Otherwise they will be automatically completed for you. Once you have pressed the 'OK' button the project will be opened and you will see a screen similar to the one below:

The quick intro map gives you a few pointers to help get you started. There is also a 'Quick Start' movie which you can launch from the Help Menu.
For information on automatically logging into a default project and on saving login details see User Administration.
For information on converting an existing MySQL database project into a local default Derby database project see Converting a MySQL Project.