/usr holds readonly system files which are not usually needed till after the initial boot process. Traditionally this directory could be shared amongst multiple machines using a readonly NFS mount. Many big packages are installed here, like word processors.

This is used to write temporary data which may last until the machine is rebooted. Often this is referred to as scratchspace. It should never be used for important or private information.

This stores readonly system files. These files are not written to by the system as part of its normal operation, but files may be changed by the system administrator as part of configuration or user changes.

This directory holds kernel state information, such as what disk drives are installed and what each of the running processes in the system is up to.

/var is indeed designed for system writing, and includes the spool directory, the log directory, and email.