I understand that Raspberry Pi boots from an SD card. What is the minimum amount of memory required by Raspberry Pi to run smoothly?
The Pi's memory cannot be increased beyond the 256mb or 512mb that board comes with. This memory can be split differently between the CPU and GPU. More info on the memory split options can be found on the elinux site, or by doing a search here for memory split.
The SD card is analogous to a hard drive. At a minimum you will need a 4gb SD card. However, you will be better of using something larger. Note that the Pi can only use up to a 32gb card and not all cards will work. You can find a list of cards known to work here (http://elinux.org/RPi_SD_cards). There are a handful of cards known to work that are larger or smaller than my recommendations, but these are exceptions rather than the rule. The smaller cards will not provide much space for additional programs etc., and the larger cards can easily cost as much as the Pi itself.
Raspberry pi performance isn't dependent on SD card size. With typical hard drive you could use it to extend virtual memory by creating swap partition, which could speed up things in some cases, but not so much with slow storage as SD card.
So the main performance factor of the SD card is it's speed which is generalized by SD class https://en.wikipedia.org/wiki/Secure_Digital#Speed_Class_Rating . Of course you are much better of buying properly benchmarked SD card ( http://elinux.org/RPi_SD_cards#SD_card_performance ) than by looking at producer claims/inaccurate class rating.
Smallest supported SD card size is usually listed on the web site you get your software distribution from. Basically, most of the contemporary distributions, Raspbian and plenty of xmbc-flavored ones require 2GB SD card, with the the only exception -- Gingerbread based Android being and just a tad bigger than 2GB, thus requiring 4GB card.
Personally, I have used 2GB card for quite a long time without any problems, however, it's usually recommended to go for a bigger (8GB, 16GB) card to significantly lower wear-off levels and increase the SD card life time.