What is server memory?

There is no such thing as "server memory"! Memory modules advertized for server usage generally use technologies which are supposed to decrease chances of errors occurring inside RAM due to various reasons such as background radiation and so on, but that doesn't mean that they are unusable on "normal" computers. You just need the right type of motherboard.

There are various techniques which make this error detection possible. The linked product uses parity and error-correcting code memory. The parity system basically works by storing an additional byte with each bit. The computer checks if the number of bits which represent 1 or 0 is even or odd and then adds another bit so that the total number gets even or odd, depending on the settings of the machine. When data is read from the memory, the parity bit is checked and if the number of bits is wrong, then computer knows that there was an error in memory storage. The downside of this is that two errors could occur and the number of bits would be right, so they would be undetected

ECC memory has some capabilities of recovering damaged data, but to do so it usually needs to store more data than non-ECC RAM and therefore may be somewhat slower. I'm don't have enough experience with it to explain how exactly it recovers data, but here's a Wikipdia article about that.

Now I'll focus on the other side of the story. RAM modules targeted at servers usually have chips which can store more data than RAM modules targeted for desktop computers and are often pushing the limits of the manufacturing technology. That, in addition to ECC, is one of the reasons why they are more expensive.

Now to actually answer your question: The computer you want is usually called workstation computer. That is name for computers that have standard desktop form factor but have performance of servers. So to get what you want, you'll need to buy a workstation motherboard or at least motherboard which supports large memory modules and ECC. In addition to supporting memory technologies common to server market, workstation motherboards often support server type CPUs, may have two sockets for dual-processor systems, are usually more reliable than motherboards for common desktops, may have large number of high speed PCI-E slots (I've seen some which only have PCI-E 16x slots for all slots on the board) and so on.

Of course, the downside of that is that both the motherboards, RAM and CPUs tend to be more expensive than common desktop motherboards.

So the bottom line is: If you need high-reliability RAM, get ECC RAM and appropriate motherboard and processor. If you just need lots of RAM, get a motherboard with lots of slots that can support large modules and use non-ECC RAM. This way you'll save more money on the whole system.