Within Teamwork Projects, you can see the current people who are on a project within the project's People tab.
You can click the 'Add People' button in the top right to add more people:
Adding people already invited to the site
In this screen:
You will be able to view all people who are already on the site, and can be added to the project. You can search for a specific name, or you can filter by role to narrow down the list of people who are listed.
People who are standard users will show as green, contacts will show as blue, and collaborators are pink.
You can click to select/deselect them individually, and you can use the 'Select All' and 'Select None' options at the bottom to bulk select/deselect. When you select someone, you will see a green tick icon to the top left of their profile image.
You can choose to include people who are already on the project by enabling the switch under the list, too.
Once you are happy with your selection, you can use the green button to add the people you've selected to the project.
Adding people who aren't already on the site
You can use the '+ Add' button at the top to add new users to your site and your project in one go. You will see the screen to add all of their new user details, you can choose to send them an invite to the site, then they will be added to the list so you can select to add them to this project.
You can also use quick add keyboard shortcuts to invite a new user, by pressing q and u.
You can also use the 'Add New Contact' button to create a new contact in the site and add them to the project, too.
Adding people from the 'Overview' section of a project
You can add users to a project from the 'Overview' section of a project by clicking 'Options' at the upper-right, and choosing 'Add People' from the menu.
You can then add and invite new users who aren't on the site, or add existing users to the project.
For more information, see: Setting user permissions on a project