< /bin, /usr/bin >
These two directories contain a lot of programs (binaries, hence the directory's name) for the system. The /bin directory contains the most important programs that the system needs to operate, such as the shells, ls, grep, and other essential things. /usr/bin in turn contains applications for the system's users. However, in some cases it really doesn't make much difference if you put the program in /bin or /usr/bin.
< /sbin, /usr/sbin >
Most system administration programs are stored in these directories. In many cases you must run these programs as the root user.
< /usr/local >
This is where you install apps and other files for use on the local machine. If your machine is a part of a network, the /usr directory may physically be on another machine and can be shared by many networked Linux workstations. On this kind of a network, the /usr/local directory contains only stuff that is not supposed to be used on many machines and is intended for use at the local machine only.
Most likely your machine isn't a part of a network like this, but it doesn't mean that /usr/local is useless. If you find interesting apps that aren't officially a part of your distro, you should install them in /usr/local. For example, if the app would normally go to /usr/bin but it isn't a part of your distro, you should install it in /usr/local/bin instead. When you keep your own programs away from the programs that are included in your distro, you'll avoid confusion and keep things nice and clean.