You actually have several questions here -- I'll reiterate them:
1. How does semiconductor memory work?
2. How can it be so reliable?
3. How can it be so dense?

All semiconductor memory is based on storage of charge (i.e. packing some electrons into a small region). If you think about any mechanism for memory, you need some way to write it, some way to read it and hopefully, a mechanism to sort the memory. It might sound strange, but most of the work of constructing a memory is in making a way to communicate to the stored information. Memory storage is divided into two regimes based on the time of retention. Memory that has short retention (say 0.02 Sec) might be cheaper to build, but it requires constant updating to keep it valid. Long retention memory is called static, and has retention times measured in years. A further distinction is made depending on whether the circuit must be on for the memory to be retained. So BIOS memory (which boots your PC) is typically non-volatile so turning the power off doesn`t clear it. The main memory in your PC (or Sega) will be DRAM (0.02 Sec retention Dram) while high-speed cache will be volatile static memory.

You question was how does it work -- I will describe DRAM since it is simple in principle. Other types share some similarity. In a DRAM, each bit of data is stored in a cell -- which is just a charge storage device (a capacitor); the capacitors are arranged into lines that are read or written all at once and the lines are arranged into planes which make a dense arrangement of the cells. Data for a particular address decodes a single line and every bit on that line is read simultaneously. In older memories, there were about 32 rows and 32 columns -- so you could store 1024 bits; in more modern memories 512x512 arrays are not uncommon, and several such planes sit side by side on the chip. Effectively, the tiny amount of charge stored in each cell on the line being read is used to generate the output value-- which is then written back to the cell (the write-back is called refresh). Each of the cells loses charge due to leakage-- which is why we have to refresh all the time.

The storage is reliable for two reasons -- the first is that the technology for making chips has been improved to staggering levels. Part of the reliability is simply due to the near perfection of the memory.
The second reason is error codes. This trick is based on having several copies of the information so that voting is possible. This can be done surprisingly cheaply-- i.e. adding only 6 bits makes 32 data safe for any single bit change.

The density issue is also spawned by the improving technology -- current technology makes transistors and wires smaller than single visible light wavelengths-- so they can't be seen in even the best optical microscopes. This makes the charge storage small as well-- but this is offest by the smaller amplifiers needed to regenerate the data.

I hope this is helpful -- you have asked a very big question. If you would like more detail or references for a particular part -- write back to ScienceLine.

It's probably far more technical than you want, but for what it's worth, there was an article on Ferroelectric memories ('smart cards') in a recent Physics Today magazine.
Aucciello, Orlando, James F. Scott and Ramamoorthy Ramesh, "The Physics of Ferroelectric Memories", _Physics Today_, vol. 51, no. 7, pp.22-27, July 1998.

The memory cards in your Nintendos are microelectronic devices; you could look up microelectronics in an encyclopedia, or in a science magazine like Discovery or National Geographic. Your library might even have a book on it. Basically it works by having tiny electrical circuits which control the flow of electricity into the computer (or play station). For example, when you advance to a new board, and the playstation needs to draw the scenery for the new board, it sends a specific electrical signal to the memory card. The memory card has electrical signals stored in it which can describe all the scenes to the computer (the same way a videotape has electrical or magnetic signals stored which describe to your VCR what images to show on the TV). When the memory card gets a signal asking for a specific scene, it sends its stored signals describing that scene to the play station. The electrical signals themselves are stored as patterns of static electricity in the memory card.