Your landline was an analog* connection to a central switch that routes traffic. If something interferes with the connection - say, a lightning strike downtown, a car passing by with bad ignition wiring, a leaking rad shield in a microwave or a ballast in a flourescent light blowing out, whatever - then you hear a little crackle or hiss in the connection and you say to the person on the other end “could you repeat that?” or you infer content from undamaged context. If another person picks up a phone extension, their voice mixes right in, and if you both talk at once, nobody understands anything until one of you shuts up.
Your data line is a digital** link, that is potentially carrying many separate streams of data - for example from bOINGbOING to your browser, and from netflix to your smart TV, and from your wifi-equipped phone to the NSA, and so forth. The data streams are broken up into packets - short bursts of digits - and if something interferes as previously described, the specific packet that was damaged has to be retransmitted. The router makes sure that the packets get where they are supposed to go - netflix to the TV and BB to your computer, and so forth.
Because you have the router there anyway, routing the packets, usually there is a bunch of additional function piled on. Some of these functions are necessary in order to have multiple devices use the router (DHCP service to assign IP addresses, outgoing address translation for example) and some are highly desirable, but not really necessary, you’d just have to be an idiot not to want them (firewalls, for example, fall in this class).
Because all the above means the router has to be fairly smart, the router vendors build them as general purpose computers, typically running linux, in order to make them affordable commodity products. This means that they are vulnerable to being hacked just like any other general purpose computer, by whoever the government wants you to believe is dastardly today (currently Russians, was Chinese, before than Iranians, before that Russians).
Did that help any?
* this means when you talk loud, the voltage gets higher, when you sing bass, the frequency gets lower, it’s literally mapping your voice from sound to electricity directly.
** this means information is being split up into individual chunks like letters, numbers, or even milliseconds of sound or video, and then those chunks are encoded as sequences of ones and zeros (+5 volts for zero, -3 volts for one, or some similar scheme). Because the chunks are being sent incredibly fast, you can send a lot of them.