OS X can only read NTFS. To write to an NTFS formatted drive you need to install third party software on your Mac. To avoid that, you can format the drive from the Windows computer as exFAT. OS X can both read and write that format. OS X can also format a drive as exFAT, but there seems to be some kind of incompatibility. Often, Windows users can't access a drive formatted as exFAT on a Mac, but have no problem if the formatting is done from Windows.
If you need the drive to remain NTFS, then you can purchase and install Paragon NTFS for Mac, or Tuxera for Mac. There used to be a paid and free version of Tuxera, but I don't think there's a free version anymore.
As far as the "incompatibility" issue, that's likely on the Windows side. By default, Windows XP does not support exFAT. You need to download and install an extension from Microsoft so XP can read/write/format that type of drive. So Windows users who say they can't read an exFAT drive are most likely on XP. So if you know the drive is going to an Win XP system, make sure they have this extension installed.