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Different crypto-currencies have different lengths of payment addresses.
For example, LTC address length is 34 case sensitive letters and numbers.
NXT address length is 20 digits.
Short payment address can be easier to remember, read, write, compare and transmit by phone, which makes it advantage of crypto-currency, making it attractive investment, doesn't it ?
The human mind can remember approximately 7 digits quickly and easily. That's how you can remember US phone numbers (sans area code) without even thinking about it. However, when it gets to 10 digits it starts to reach the limit of working memory, and 20 digits is definitely out for most people. Of course, if you actually spent the time to try to memorize it, 20 digits is a lot easier than 34 alphanumeric characters.
That said, I don't think the benefits justify the costs. Cryptocurrency is designed to be used by people with access to a computer or smart device. Copy-pasting is a lot easier than trying to remember a bunch of addresses. Granted, it might be useful to remember one address, your own, that you can give out to people in conversation. But these days most people have their smartphones on them wherever they go. Also, regarding "transmit by phone": Only if you use a touch-tone phone. Otherwise, you can just copy-paste and then SMS/email/submit it to the other party.
What disadvantages are there? Well, now that there are only 10^20 addresses, or 66 bits - much less than the 160 bits of Bitcoin and pretty much all of its clones, decreasing security. Moreover, this also means addresses must be generated on-network to prevent the possibility of duplicate addresses and registered as an account. For NXT, only 12 billion addresses are necessary for there to be a 50% chance of collision (a la birthday paradox). For Bitcoin, 1.4 septillion addresses need to be generated for that to be the case - which for all practical purposes, will never be generated. So people can generate addresses without being connected to the network and be very confident that no one else has ever generated the same address, making cold storage solutions easier, for instance.
Edit: To answer the question in the title: I only know of two address schemes, Bitcoin-like and NXT. For Bitcoin-like addresses, the first character is always something which identifies the coin (not uniquely - multiple coins, including Bitcoin, begin with a 1, Dogecoin and Digitalcoin both begin with a D, etc.), followed by a 160-bit hash and a checksum. Then you have NXT, 20 base-10 digits. I don't know of anything shorter than NXT, and any shorter would be unreasonable given the increasing probability of collisions.
