You can purchase a membership on the behalf of a constituent by navigating to the General Membership Module > Membership grid, and hovering over the dropdown menu that reads +New. 

In the new dialog window, select which membership page you would like to open. This will open the membership page and you can enter the member information as needed.

If the constituent already exists in your database, you can also go to the Membership Tab in their record. Click on the green Add Membership button to select the membership page. From here, the constituent's contact information will automatically populate on the page. If renewing a membership, select the row with the membership to renew, and a new button to Renew Membership will be revealed. It also allows you to select the appropriate membership page to complete.