Terminology
Connection
Connection defines which JIRA instances will be synchronized. You provide here a remote URL, Technical User, etc.
There is only one connection between two JIRAs defined.
Contract
Contracts are bind to connection. Contract is telling WHICH issues and WHEN will be synchronized. For example, Bugs from "Test Management" project will be synchronized with provider's JIRA.
Contract contains Fields Mapping.
Fields Mapping
Definition of fields that should be synchronized for example: Summary, Description, Priority, Due Date.
Mappings can be defined as a Field Mapping Templates.
Triggers
Triggers tells when issues should be synchronized. That included Create, Update events list as well as Comments and Attachments in different flavors.