What is it that makes Bitcoin unique amongst crypto-currencies? Is it just the most popular one or does it have better design aspects too?
Until a careful analysis is performed, I think most crypto currencies are very similar in all properties except one - adoption.
Bitcoin has the largest hash rate, participating nodes, accepting sites, and people going over the code and protocol.
While alternative currencies can gain more hash rate, especially if effective forms of merged mining are developed and adopted, it remains to be seen if they gain the same adoption (sites accepting them), and developer base (strong developer community).
(See this post about merged Namecoin/Bitcoin mining)
Edit - One notable exception to this is Namecoin, which does offer unique features making it suitable to use as a DNS or general name registrar.
From a theoretical point of view, the interesting part of bitcoin is that it lacks a central issuing authority (often called a mint in the crypto-currency literature). In terms of security an anonymity of users, it does not have particularly strong features. Refer to answers on the following two questions for more info:
The thing that separates Bitcoin from other cryptocurrencies is the amount of computing power the network has. It is extremely difficult to 'overtake' the network because it would be costly.
I wouldn't say it has the 'best' design aspect to it as something catastrophic as a price crash would cause miners to pull out. It might take months or weeks for the difficulty to get back on par. Transactions will be super slow, as there are few miners to solve the blocks.
Something like SolidCoin fixes this problem, I really like the twice daily difficulty adjustments as it prepares the network for something Bitcoin might be prone for.
Brief comparisons of the different alts here: