Phone Calls

"Under Egyptian legislation the authorities have the right to issue such an order and we are obliged to comply with it," Vodafone Egypt explained in a statement shortly after. Along with Vodafone, Egypt's other three major ISPs, Link Egypt, Telecom Egypt, and Etisalat Misr, all stopped service. Jim Cowie, the co-founder and CTO of internet monitoring firm Renesys, told the WSJ:

What is most likely is that somebody in the government gives a phone call to a small number of people and says, ‘Turn it off.' And then one engineer at each service provider logs into the equipment and changes the configuration of how traffic should flow.

It was likely as easy as that.

BGPs

Renesys saw the effects immediately. Some 3,500 Border Gateway Protocol or BGP routes—the places where networks connect and announce which IP addresses they are responsible for—disappeared in an instant:

At 22:34 UTC (00:34am local time), Renesys observed the virtually simultaneous withdrawal of all routes to Egyptian networks in the Internet's global routing table. Approximately 3,500 individual BGP routes were withdrawn, leaving no valid paths by which the rest of the world could continue to exchange Internet traffic with Egypt's service providers. Virtually all of Egypt's Internet addresses are now unreachable, worldwide.

But Stéphane Bortzmeyer, an IP communications whiz, surmised that Egypt pulled the plug on the net literally: "BGP is the symptom, not the cause. The cables have simply been unplugged."
Withdrawing BGP routes (or just unplugging cables) is a much more effective way of blocking the internet than, say, turning off DNS, in which case users could use DNS from overseas to access the internet. Compared to Tunisia, where certain BGP routes were blocked, or Iran, where internet connections were simply throttled, Egypt's disconnection is a severe one.

Disconnected (Almost)

As of last night, Renesys estimated that 93% of Egyptian's networks were unreachable, with only one service provider, the Noor Group, still serving its customers. It's unclear why they're the only ones who didn't get turned off, though some are speculating that its role as service provider for the Egyptian Stock Exchange is what's keeping it online.
Reports from Egypt suggest that citizens may be able to use dial-up to access the internet, and LifeHacker has the nitty gritty on how to do it. It's not going to be fast, but it seems like for a vast majority of the Egyptians, it might be the only option. [Renesys, DomainIncite]