What piece of hardware directs traffic between the connected devices when two LAN's are communicating across the Internet?

The correct answer is "Router".

Explanation:

In networking, routers is one of the main hardware which is used to direct the data traffic on a LAN system.

The router also works as an interconnection system for two or more than two networks. Routers mostly work between the internet and computer which directs the data to its right path.

It works between the computer system in a premise to make the networking possible through a LAN by using switches for communication between two or more than two devices in a LAN across the internet.

Hence, the most appropriate answer is the router.