Take Command uses only documented Windows APIs to access the file systems, so it works with any file system supported by Windows.
A network file system allows you to access files stored on another computer on a network, rather than on your own system. Take Command supports all network file systems which are compatible with the underlying operating system. The networking software used to access remote systems (such as UNIX, Linux, OS X, etc..) which use different file systems typically emulates one of the common Windows file systems. Those emulations do not always provide a perfect duplicate of some functions (attributes, timestamps, etc.), an issue unrelated to Take Command.
File and directory names for network file systems depend on both the "server" software running on the system that has the files on it, and the "client" software running on your computer to connect it to the network. However, they usually follow the rules described here.
Most network software maps unused drive letters on your system to specific locations on the network, and you can then treat the drive as if it were physically part of your local computer.
When you use a network file system, remember that the naming rules for files on the network may not match those on your local system. For example, your local system may support long filenames while the network server or client software does not, or vice versa. Take Command will usually handle whatever naming conventions are supported by your network software, as long as the network software accurately reports the types of names it can handle.
In rare cases, Take Command may not be able to report correct statistics on network drives (such as the number of bytes free on a drive). This is usually because the network file system does not provide complete or accurate information.
Some networks also support the Universal Naming Convention, which provides a common method for accessing files on a network drive without using a mapped drive letter. Names specified this way are called UNC names. They typically appear as \\server\path\filename, where server is the name of the network server where the files reside, and the path\filename portion is a directory name and file name which follow the conventions described under Directories.
Take Command also allows you to use UNC directory names when changing directories (see Directory Navigation for more details).
Take Command and TCC have built-in support for OpenAFS. The parser will recognize Linux-style AFS names (i.e., /afs/athena/user) and convert them to Windows-compatible names (i.e., \\afs\athena\user). (It will also check for custom AFS mount points, and use that name instead of afs.)
See http://www.openafs.org for more information on OpenAFS.