Internet infrastructure through Analogy.
A Series of Tubes
“A series of tubes” entered the collective concious as soon as the octogenarian senator from Alaska used it. For some, it's a way of conceptualizing the way that data flows through networks by way of analogy to older technology and infrastructure; to others, it symbolizes the technology divide between generations.
It isn't not apropos. In context, the statement was designed to explain that Internet service providers had limited bandwidth—by ways of limited infrastructure—with which to transport data around its network. The Big Inch pipeline between Texas and New Jersey provided for twice the flow-rate (bandwidth) as previous pipe-lines given the increase in pipe diameter from twelve inches to twenty-four inches. Likewise, sending data between two connected locations on the Internet can be doubled by doubling the size of the proverbial pipes, e.g. by doubling the number of fibre connections. Nevertheless, it's been met with derision by a generation that feels at home using the technology—so called digital natives.
The Internet started as a series of big computers (hosts) that were frequently connected to each other. Users would often dial into one of these machines, and then could reach the rest of the Internet from there. This was later replaced with personal computers connecting to ISPs and providing direct access from their computer to the Internet.
The phrase direct access in this case is a little misleading. While users connecting to services like Prodigy or NetZero could reach any other machine on the Internet, connections would only have the appearance of being between the two end-points (the End-to-end principcle). In reality—and transparently to the user—these connections operate the same way they used to, with the user's computer talking to the remote service by sending data first to the user's Internet provider which in turn sends it closer to the destination and so-on through various intermediary computers.
The analogue to old-tech is a fire-fighting bucket brigade, where one party is the water source and the other the blaze. Like the various computers required to move a packet from a source to a destination, the people in the brigade move a bucket from the source to the destination by receiving it and handing it off to the next person towards the destination. The beauty of these situations is that nodes along the way only need to know the next person in the line, and not the full path necessary to reach a desired destination.
The bucket brigade functions as a transport method. It's concerned only with providing a way of getting something from a source to a destination, and if some water (or a packet) were to be lost, it will be resent from the original source rather than ensure that every drop makes the journey. In technical jargon, this is called best-effort delivery. (If delivery guarantees are necessary, the source and destination need to build this robustness on top of the best-effort system.)
Analogues aren't perfect, especially when they involve the digital world. A bucket brigade functions by moving a physical bucket with physical water, and the movement of the bucket by the laws of physics ensures that if it reaches its destination it's no longer at its source. It can only be at one place at any one time, and any spillage (or siphoning) is visible if the volume at the destination is compared to the volume that left the source. In the case of a digital network, movement only happens between any two nodes; once a node receives a packet, it makes a second copy which it sends on to the next node in the brigade. Normally, all local copies are then destroyed after it has been sent (forwarded) to the subsequent host.
The other copies don't need to be deleted, however; nor would there be any evidence if rather than forwarding the message on to just one destination, a host sent a second copy to an additional destination unbeknownst to any other member of the brigade. In 2006, an AT&T technician revealed first hand knowledge that the National Security Agency had three floors in an AT&T's building that captured all data going through the AT&T network in that building and the understanding that other network junctions around the country had similar rooms.
Information Super Highway
Long before “a series of tubes” the metaphor of the day was “the information super-highway”. Road networks like the Internet are designed to get from point A to point B in an efficient manner by creating a hierarchy of connections. The lowest tier in the hierarchy works to connect to end-points: allowing a home to connect to the road network or onto the Internet. Neighborhood roads connect to local thoroughfares which eventually connect to larger roads and eventually to national highways designed for high-bandwidth users. Similarly, users connect to their local Internet access (service) provider which connect to a larger tier-2 network and eventually provide access to remote destinations by connecting to high-speed links of tier-1 backbone networks.