A partial membership is when a contact has an outstanding membership (usually up for renewal).
Any Partial Membership can be viewed at Memberships > Reports and select either "Active+Partial" or "All" and you will see next to the contact names on the right side a green button with 'Active' and the blue 'Partial' button (there are also Cancelled and Expired status).
It gives you an idea of who has an outstanding membership and a simple email reminder can be sent to the individual.
Partial memberships will also be seen within a Contacts profile.
Partial memberships will expire the same as an active membership - it's up to you to chase down the extra funds, or cancel the membership if they don't pay the balance.
We don't currently handle any sort of grace periods, you can of course manually override or add people to events where member only tickets exist etc.
A real world scenario for Partial memberships are primarily when people make part payments on their memberships, they don't become a fully Active member until they are completely financial.