The /bin directory is where executable files are kept. all users can use the files in the /bin directory.
The /boot directory is where information needed at boot time is kept. Important files such as the Kernel are kept here under the name of vmlinuz or zImage. Not all systems use the /boot directory but it is becoming more common and accepted.
The /dev (device) directory is where linux keeps the device files. Linux treats everything like a file including devices such as your floppy disk drive, printer, modem and so on. When you write to your floppy drive you are actually writing to the /dev/fd0 file. There is a file called 'null' in the /dev directory and anything sent to it will be deleted instantly. This little file can be quite handy but be careful using it because once the data has been sent to 'null' it is deleted and gone for ever.
The /etc directory is where linux keeps its configuration files. By convention configuration files end with .conf or .rc
This is where a users home directory is kept. Every different user will have a different /home directory. When you log in as a user (which is a good idea while you play around with linux) you will go to your home directory after logging in.
The /lib directory is where shared libraries and links to libraries are kept. Shared libraries are used by many programs that require the same piece of code rather than each program using up more memory with what essentially is multiple copies of the same code they simply use the shared libraries.
The lost and found directory is where Linux will keep any files it restores after a system crash. Usually you will have no problems with Linux but if your system ever crashes and you loose files try looking in the /lost+found directory for them and you may be lucky enough to find them intact. This may save you a lot of trouble one day so it is good advice to remember.
the /mnt (mount) directory is where all the external file systems are mounted. For example in order to use any other file system such as a CD-ROM you will need to mount it first. You can mount a directory anywhere you like on your system but keeping all the mounts in one directory makes life a lot easier considering you will need to mount any external system including floppy drives, CD-ROM's, other hard drive and any network storage devices you can imagine how confusing it can become if your mounts are not all in the same directory.
The /proc (process) directory contains all the 'virtual files' representing different aspects of your system such as your cpu, RAM etc. You can go through this directory and obtain information about your system using the cat command. You should also remember that the /proc directory is relatively a new addition to the Linux world and may not be present on some older versions however most system now employ the /proc directory as a valuable resource.
The /root directory is similar to the /home directory except it is the root (super user) user's home directory.
The /sbin directory is identical to the /bin directory only it contains executable file that only a super user (root) can use. You will need to be logged in as root to use anything in the /sbin directory.
The /tmp (temporary)directory is used by programs to store temporary files. You will find a file named install.log in the /tmp directory and if you view it you will find a log of the installation (handy to have).
Originally the /usr (user) directory started out as a 'user directory' where user related programs and files were stored. It has evolved to also contain read only files that you don't need root access to view and also to hold X-based programs. The /usr directory has evolved to become a directory of mixed use though it is generally a place to store a users files that may be executed, read and/or (though not necessarily both read and write privileges) used freely without root access.
The /usr/local directory is an optional directory where a user stores and installs any applications they use and or download. The idea is that this directory keeps things organized and keeps user programs away from the main (often more important programs). It is also used to store downloaded material from the internet so it can be easily located at a later time.
The /var directory is where files that may change over time are stored. Some of the things found in the /var directory include spools for printers and mail. Also found in the /var directory are log files indicating various logs of different system activity.