"Syncing" (short for "synchronizing") is a process of reconciling data shared between a Palm and a desktop machine. Information entered on the Palm since the last sync is replicated on the desktop, and vice versa. This helps to ensure that even if something catastrophic happens to the Palm, little data is lost.
There are several ways to hook up a Palm to a desktop in order to transfer the data. The original Palms used a standard RS-232 serial link, and later an Infrared link. Modern Palms use USB, Ethernet, or even wireless connections like 802.11 and Bluetooth. Linux supports all of these methods.