The distributed nature of the internet makes it really impossible to destroy all of it, and all the information on it, without destroying every computer as well, which you seem to be excluding. To erase all access everywhere you must contact every computer on the internet.
In particular I need to ask what you define as 'the internet'. You have a rough hierarchy of computers connected, how many interconnected computers is 'the internet'. Is an intranet at a large company still 'the internet'? What about an intranet of a small company all within one building? My connecting my three computers at my house into a tiny intranet? Is an internet that can connect to all clients hosted by my local ISP only, but not make the 'hop' to a higher ISP (meaning can't reach beyond my city) count as internet? Depending on how loose your definition of internet is it can be nearly impossible to take out the internet (I plugged my cellphone into my laptop. That's two computers communicating, it's the internet!)
I don't believe any of the posted answers so far are viable ways to take out, or even more than mildly inconvenience the internet, but that isn't the answer's fault, it's nearly an impossible challenge without an apocalypse that destroys all physical computers.
However, there are a few approaches that I can see which you could use to varying degrees, depending on how much you need to actually destroy the internet for your story:
Remove access to large portions of the internet easily
note, this solution also removes phone, cable TV, and most other forms of electronic communication in addition to internet
It's not that hard to isolate parts of the world from large parts of the internet. Right now data going between continents travels through a few dozen wires spanning the ocean. Not only is the number of lines relatively small, they are completely unsupervised, you can mess with them without anyone seeing you.
Cutting a cable is not entirely trivial, but also hardly impossible. You would need something capable of diving deep enough into the ocean to get to the cables, with sufficient cutting power to cut through cable and the protective encasing. This is more expensive then a regular individual can manage, but even a small nation can probably get together enough money to get a cable-cutting sub built. If you don't want to destroy the infrastructure, tricks to disable the communication without fully cutting the wire surely can be made easily enough once you reach it.
People could repair these connections, but as long as you have a cable cutting sub you can cut new connections as fast as they can be repaired; it's nearly impossible to watch all of line from the US to Europe at one time, it's just too long to properly watch for malicious attacks against it.
Imagine Australia if the internet lines to it were cut. Most hosting for large companies and services are located out of Australia, they would lose their Amazon and Wikipedia and many other major sites they use regularly. They would still be able to communicate with anyone in their home continent, and in fact many major businesses may have servers in Australia for latency reasons that would still work, but the internet would be a fraction of what it used to be. For all intents and purposes Australia will be missing most of the power of the Internet.
Other places will do better. The US has the most internet infrastructure of any other nation, so many of the big names will have servers within the US. Anyone in Europe will have access to any server on any landmass connected to Europe, which is the majority of all land, so they likely will also not suffer as much; However, both the Americas and Europe/Africa would suffer from lack of connection to the other major land mass. The UK and Japan, two first world nations who make huge contributions to the internet in general, will suffer significantly since their small size means relatively small number of major services have servers within their nation.
People within these nations would still be able to use satellite internet, but considering the small number of satellite and the huge number of people there would be significant bandwidth issues, to the point of making the internet effectively destroyed until the countries figure out a way to regulate limited satellite bandwidth in some manner. In theory a country could also try to take out satellite as well. This is harder then taking out the physical lines, but not entirely impossible.
Take out HTTPS
This probably isn't what you want, but removing security from the internet will significantly weaken its usefulness, if not destroy all of it. Right now our secure internet is dependent on two things:
- No algorithm can decode keys as efficiently as we can encode them
- We trust our top level Certificate Authorities
If either of these presumptions were proven invalid security falls apart. Suddenly with minimal effort one can set up a man-in-the-middle attack and get all my banking information, taking my money over night. They can also inject misinformation or viruses into any request I make online, since they can simply replace messages with whatever they want me to see. This would not destroy the internet, but it could have horrible repercussions.
Getting access to the private key of just one CA would allow quite a bit of evil, but publishing an algorithm that allows decrypting of a message using only the public key would truly break security since all CA are equally invalidated at once.
The internet knowledge still exists with this approach of course, it's a limited solution to what you want at best.
I added this option, knowing it's quite different then your stated goals. However, I think that, while not breaking the internet, this has room to have a significant affect on internet, economy, and more. I've asked a question inspired by the idea just now, perhaps it will give some ideas of how much harm something like this could lead to: How can a small country exploit breaking HTTPS to destory a larger country while growing its own strength?
Big bad scary virus
The other option to get closer to what you want is a virus. Of course I don't consider it at all realistic that a virus would actually be able to take down the internet, but you get some poetic license here. If you truly want all the internet to go away I suggest using a hand wave virus without explaining exact details and ask your readers to give you some suspension of disbelief; because any answer that does what you want is going to require quite a lot of it. Ultimately it's better to give an answer where you don't explain some details than to try to explain everything and, in so doing, make it obvious to informed readers just how improbable your suggestion really is.
There are some things you can do to make the virus idea more believable, to at least get a little leeway from readers.
Imply that someone utilized some 0 day exploit (i.e. found a bug no one knows about and exploited it before anyone knew it existed to defend against it) in the very infrastructure of systems that meant that you could spread a dangerous 'virus' in such a way that anyone trying to connect to different services will likely contract it. Basically, someone found a huge security hole that nobody realized was there and exploited it to make an uber-virus before anyone could defend against it.
Usually, protecting against a virus after it's released isn't that hard - people will find ways of blocking it and spread that information - but they spread it with the internet. Thus, if a virus did worm its way out there and knock out the internet in one go, the method for identifying its root cause, creating protective measures, and spreading them to others would be greatly limited. The virus could also be destructive (for instance, permanently removing data from infected systems to render the information inaccessible).
I would go on to say that the virus was subtle in the manner it was crafted. Someone released the virus a while ago and gave it time to spread and infect most ISPs before activating it. In fact say they also infected the certificate authorities and got hold of their private keys. It can now do a perfect man in the middle attack, getting through SSL security by pretending to be your security authority. Whenever you attempt to hit a website it can instead send its own virus back to your computer to spread further.
Suddenly all internet traffic is disabled because your ISP won't forward your requests, and many devices attempting to connect to any site anywhere are instead infected by the super-virus. Everyone is now too afraid to go online or have already had their devices infected, and depending on how nasty the virus is even after new ISPs are built people will be afraid to go online until they find ways to fully protect against the virus and stop it from spreading to the new ISPs via infected systems that connect to it.