Here is an analogy for you:
HD=Well,
RAM=Water Tank
Programs=People
Say you have a nice big well, but it is 2 miles away. Closer, you have a small, easy access water tank. With only 5 people in your village, the water tank is never used up in the day, and you never have to make the slow journey to the well. One day, 10 new people move into town. Now the water tank is too small. When everyone goes to get a drink, you run out of water and must make the trek all the way out to the well. This is very slow.
Now you decide to increase the size of your tank. No longer do you have to run out to the well when everyone wants a drink. You can go to the quick and easy tank and never have to worry.
In conclusion:
Having more RAM prevents your computer from having to use the slow virtual memory on your HD. It will always be able to use the very, very fast RAM to acces program data, rather than the virtual memory. Even when many programs are running, you never have to deal with the slowdown of using virtual memory.
Is my analogy clear? My technophobic mother found it very easy to understand.