In particular, I was pool-mining with scrypt algorithm, using my GPU (was not bitcoin, but another coin named gamecredits, at gmc.acidpool.com).
I would like to purchase cheap mining hawdware (which would be useless with bitcoin but perhaps good with relatively "new" coins). Should I care whether a hardware supports "scrypt" in this case? Or does hardware care about a more "primitive" level without caring about the algorithm?
A GPU is general purpose and can do many types of computation. An Application Specific Integrated Circuits is completely specific to one application, they don't have any ability to do any computation which they aren't not designed for. A mining ASIC made to perform scrypt hashing for example can't do any other type of hash.
Different currencies have different mining algorithms. What you should do is to use coinwarz.com to calculate the most profitable one, by filling in the details of the mining hardware's specs of the miners you're thinking about buying.
Be sure to put cryptocurrencies' future values (predict them!) and when the next miner may be released (predict!) into account (which coinwarz can't)
