What does it mean for a cryptocurrency to be ASIC-resistant? I guess the question I'm asking here is what does it take for an altcoin to be ASIC-resistant? What does it need to have?
It means that there will be no significant speedup by implementing the algorithm in an ASIC, as compared to a CPU based implementation.

This is usually achieved by requiring a lot of memory, which when implementing this on an ASIC, translates to needing lots of physical area on the chip.

ASIC implementations derive their power from having many physically parallel and pipelined threads on one chip, each requiring a certain piece of chip real estate.
Now if your algorithm requires lots of real estate even for one step of the pipeline, there will not but much room to actually parallelize or pipeline the algorithm.  Thereby you will lose the parallelizing advantage of an ASIC implementation.

scrypt is such an algorithm. Used for instance by Litecoin.