How to trigger synchronization

Create	You can choose when/on which JIRA events trigger remote issue creation. Most common event for synchronization setup is ISSUE_CREATED - when you create an issue in local JIRA instance, remote one will be created as well. You may use a workflow that at certain transition triggers some event. That event can create remote issue too.
Update	You can choose when/on which JIRA events trigger remote issue update. Most common events for synchronization setup are ISSUE_UPDATED and ISSUE_COMMENTED. Issue must be created first to make update happen! If it is not, there will be no synchronization.
Comments	See Comments & Attachments
Attachments	See Comments & Attachments
Workflow Mapping	You can select corresponding workflow mapping. From now on workflow will synchronize, so when you make a transition in local JIRA instance, remote issue will be transitioned as well (corresponding transition configured in workflow mapping scheme in remote JIRA instance) See Workflow Synchronization

Trigger section is split in two: local and remote triggers. (It will be up to date after configuration is synchronized. It is done automatically in the background. May have some delay. You can force it by pressing "Synchronize configuration" at Connection.)
So you may know what to expect from remote JIRA.

Triggers can be:

Automatic (Events)
Manual (Create Remote Action)

JIRA Events can be thrown in different scenarios. The most common are built-in events triggered by system and workflow events triggered by transitions. You can also use programmed events (add-ons or groovy scripts).

Mind that issue must met JQL constraints to make IssueSYNC listen for events.