fat or fat16 is what a floppy disk uses and what older hard drives have to use. Hopefully you won't worry about this one much anymore. But it is the basic level of fat32.
Windows 95/98/Me act basically the same except for some appearances and addons, so when I say Win9x for most fat32 based I mean all of them. Win9x uses fat32 which Linux and NT/XP/2K can both use as well as its native filesystem types. Linux uses ext2 and ext3(RH7.2) filesystem where NT/XP/2K uses NFTS and NTFS5 (XP), which 9x can not use any of them.
Since XP is new newest version of "NT" I will use it for use when speaking of the NTFS filesystem. I won't even try to tell you what each one is so here is a link if you really want to know. For my and your purposes you just need to know that Win9x can not read NFTS partions. Linux can read-only with some effort and writeable with some knowlegde and finess. In Windows 2K and up you have a disk management section that will some do wonderfull things and damaging things.
Linux uses the ext2 filesystem and ext3 as of 7.2, forget if in 7.0 and up or what. Except for the swap, Linux can install on a fat32 partion as far as I know, never did it though. So that would mean it read and writes with no problem, yes. It will read NTFS with some work and write with alot of work. Though I would like to recomend using a small fat32 to share files between the two, of course this affects the drive letter assginmnets of all devices except c:\
I didn't want to go into the differences between these variuos filesystems due to the wide availability of content on the subject already. I am just giving you the basic knowlegde that they are different.