There is an important distinction to be made between "routing" and "forwarding." Briefly put, "forwarding" is the data-plane activity of taking a L3 packet in on an interface and (usually) sending it out some L3 interface. However, "routing" is the control-plane activity of deciding how to get to a certain L3 destination. This distinction is so important because people often use the term "routing" or "router" to describe boxes that basically don't and aren't.
In my personal opinion, static routing alone does not count. That is to say, a device that runs no routing protocols or has any other smart metrics to make routing decisions is never actually making any routing decisions; they have all been made for the device by the device's administrator. If you consider static routing to be routing, then every desktop system you have is also a router, because it uses a static routing table to decide where to output packets and what the destination L2 address needs to be. Since I don't consider your desktop PC to be a router, static routes don't count.
Now, that said, you can see that most SOHO devices don't qualfy as routers, because, again, they don't make any routing decisions at all. They have been explicitly configured with static routing information and that's it. They typically have an unusual forwarding path that is geared towards port address translation (PAT) rather than pure IP forwarding, though some allow you to turn the "firewall" PAT off and get a normal IP forwarding path.
As a matter of pure routing protocol bigotry, RIP does not a "real" router make. Yes, technically a device with a forwarding path and RIP would be considered a router, because it can make routing decisions. But because it's using RIP to do it, I would never be confident in it making the *right* routing decisions. So I consider that case a toy router. You will see some SOHO devices and even entry-level business devices that support RIP, primarily because they think that makes them a real router. But they're wrong.
So in my personal opinion, a "real router" is a device that has an L3 forwarding path (e.g., IP forwarding) and also makes its routing decisions using one or more useful routing protocols (e.g., OSPF, IS-IS, BGP).
L3 switches are a special case to consider. Most L3 switches do not have full hardware L3 forwarding paths, only hardware cache-flow engines. They do, however, do implement a full L3 forwarding path in software, which populates the hardware cache. By my definition, L3 switches are in fact real routers (if they implement real routing protocols, too), but they have architectural limitations that make them only appropriate in certain situations. The same goes for Cisco's NSEs.
Now that's slightly different than when you see hard-core networking guys who talk in chest-thumping terms about how they use the biggest/fastest. When you're having that kind of argument, where "real router" means biggest and the best and evolves over time, that's not at all a technical term and can't be defined. (And anyone who says that the AGS+ isn't a real router, well, them's fightin' words)
Cooky, IMA is inverse-multiplexed ATM, usually where they use multiple T1s as a fat ATM channel. Because ATM is Evil Incarnate(tm) you need not worry about it. This is not a technology you want to use.
FDDI is dead because the chips haven't been available for a few years now. FDDI did rock hard back in the day, though. I know some folks who are trying very hard not to give up their FDDI but will be forced to convert as their spare gear stockpiles run out.
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