The correct base url for the hosted GitLab is https://gitlab.com/api/v4/ so your request to
GET /projects would be

curl --header "PRIVATE-TOKEN: XXXXXX" "https://gitlab.com/api/v4/projects"

That would return all projects that are visible to you, including other user's public projects.

If you wish to view just your projects, then you should use the GET /users/:user_id/projects endpoint, where :user_id is your user ID that can be found on your GitLab profile page or in the response to your request to GET /user if you're authenticated.

# Get :user_id from this request
curl --header "PRIVATE-TOKEN: XXXXXX" "https://gitlab.com/api/v4/user"

# See your projects by replacing :user_id with id value from previous request
curl --header "PRIVATE-TOKEN: XXXXXX" "https://gitlab.com/api/v4/users/:user_id/projects"

Also, the project ID is not the same as the project name. You can retrieve the project ID from the response of your request to GET /users/:user_id/projects, or from the project's settings page.

I just found out an even easier way to get the project id: just see the HTML content of the gitlab page hosting your project. There is an input with a field called project_id, e.g:

<input type="hidden" name="project_id" id="project_id" value="335" />

The https://gitlab.example.com/api/v4/ is API address of your GitLab instance (if you are using GitLab.com, choose https://gitlab.com/api/v4/ instead) , and /api/v4/ is a prefix and should be used for all scenarios!
GitLab API provides so many resources, see: GitLab API

Some examples:

You need to get your project's releases (Projects API):

curl --header "PRIVATE-TOKEN`:` ${YOUR_TOKEN}" http://gitlab.example.com/api/v4/projects/${PROJECT_ID}/releases/

You want to create a new Issue (Issue API):

curl --header 'Content-Type:application/json' --header "PRIVATE-TOKEN`:` ${YOUR_TOKEN}" --request POST "http://gitlab.example.com/api/v4/projects/${PROJECT_ID}/issues?title=${ISSUE_TITLE}&labels=${LABELS}&description=${ISSUE_DESCRIPTION}"