The ODN is the Optical Data Network, and runs alongside the EPS (ElectroPlasma System) to all computerized systems on the ship.
Every control panel, light fixture, and warp nacelle has an ODN connector, which the computer uses to command and control the device.
When Geordi says that he's working on the "Main ODN Line," he's referring to a trunk line which travels from the main shipboard network to the nacelles. This would collect data from the nacelles and dump it on the network, and collect data from the network and dump it to the nacelles.
The problem here is that it's now a single point of failure; the point of the ODN was to have multiple pathways for data to travel around the network. For example, there was a direct line between Main Engineering and the Engineering station on the bridge (TNG The N'th Degree), and likely many more direct access links to avoid network congestion.
One thing that many people don't realize, because it's not brought up much on the show, is that the Enterprise doesn't just have one Big Iron Computer mainframe. It has three mainframes, all linked to each other, running inside warp fields to process data faster, and many hundreds of subprocessors and microcomputers all linked to each other via the ODN. This required MASSIVE amounts of bandwidth to coordinate. (Star Trek TNG Technical Manual)
Two more things:
First off, it appears that the ODN carried data with high powered lasers, or else the sheer amount of data being pumped caused the fiber optic cables to carry a huge amount of energy, as ODN relays and lines were known to explode, not usually considered a failure mode of optical fibers.
Secondly, there was also a radio-based network available used for communicators and PADDs, kinda like Wi-Fi, many many years before Wi-Fi was ever a thing. In TOS, communicators were like telephones; you called into the Communications station/department, and had your call routed, or you could select a destination on the intercom. Starting in TNG, all communicator communication was point-to-point over the radio network; this network not only connected mobile devices inside the ship, but also from ship to shore and ship to ship, with automatic call routing. This radio network could also be used as a backup for the ODN, but it didn't carry nearly as much bandwidth as the ODN.
Interestingly enough, this implied "radio data network" is only mentioned in the Technical Manual and in Star Trek TNG novels... I can't think of any on-screen appearances of anyone using the radio network as a backup or as a component of some problem solving technobabble.