Connection

(1) 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

(2) 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.