Citation from arduino home page:
"Arduino is an open-source electronics prototyping platform based on flexible, easy-to-use hardware and software. It's intended for artists, designers, hobbyists and anyone interested in creating interactive objects or environments."
Arduino is microcontroller open-source prtotyping platform. There is a lot of arduino type boards. From very small arduino nano across basic arduino Uno to arduino Mega or Due. All schematics of all types are for free, so everyone can make one for himself. Also development environment for arduino (arduino IDE) is for free, you can find it on home page of project. Its syntax is similar with C programing language.
Arduino board has few analog input pins and a few more digital pins, which can surves as input or output.
After driver installation, it is possible to connect arduino directly to PC with USB cable and communicate with it like it is a serial port device.
Shields serves as extensions for arduino board. It extends arduino functionality. And there can be found planty of shields on market. One can even use more shileds at once, but it is necessary to be carefull witch pins are in use by the shield and which are still ready for another usage.
Source of pictures: robotshop.com
Meaning of libraries is same like in another programing language. It is a piece of code useable by more programs. Their purpose is to save programmers work. All libraries for arduino are for free and free to share. Thanks to arduino popularity, there is a lot of available libraries, so if you want to use some shield or modul for arduino, you can be sure that there is a library which can make all work much more easy.
MS Windows Easiest way is to copy directory containing library files (directory must have same title as files .cpp and .h in it) to directory with libraries in arduino home directory (e.g. C:\Program Files\Arduino\libraries\). And restart arduino IDE.
Linux You can use same style solution like in windows, just to copy the directory with library to directoty with libraries in arduino home directory (e.g. /usr/share/libraries/arduino/) and restart arduino IDE.
In principle exist a lot of ways to do it…
You can find some informations on this topic on adafruit pages.