A 12V trigger is used by accessory items that are powered by 12 VDC. Such items would normally be motorized projection screens or lifts for TV's or speakers. The trigger is merely a device to signal a "turn on" to the connected item.
"IR in" normally accepts an infared remote signal from a dedicated remote control, normally one made by the same manufacturer as the receiver/processor. "IR out" lets you daisy chain remotes throughout a whole house installation.
Depending on the equipment, most users never connect to either of these jacks and instead buy dedicated pieces of automation from a company which specializes in just that sort of equipment. At the high end of the market, infared remotes are usually replaced by UHF models.