The Assessment Library is the place that houses all pre-built assessments that have been shared by the Edulastic community as well as the master copy of any assessments you have created yourself. These can be searchable by using either the keyword search box on top or the filters on the left hand side.
It also houses those assessments that you have created. They can either exist in draft mode(where it has not yet been shared) or in published mode(where it has been shared either at the school or district level, in the public domain(for all Edulastic users) or kept privately). Once you assign an assessment, it then moves over to the Assignments tab.
Assessments that you have created can be found by clicking on the Authored by Me button: