ExFAT is an updated version of FAT (Microsoft DOS's original File Allocation Table) meant for use across platforms (not only PCs) and on external drives. Don't confuse exFAT with EXtFS. ExFAT is still a Microsoft product, meaning companies (hardware companies, like memory card manufacturers, who want to use exFAT on their disks/drives) need to pay a licensing fee to Microsoft. It is not open source. ExFAT is the file system I recommend for external USB disks.
Fortunately for Mac users, Apple has an agreement with Microsoft and so OS X supports exFAT drives (as of Snow Leopard). You can plug in an external exFAT USB drive and read and write to it on your Mac. You can take an existing disk and reformat it as exFAT right from OSX. There is one caveat: the disk which you format from OSX as exFAT may not be readable by Windows machines!! So either test a newly formatted exFAT drives on a Windows machine before using, or format it from a Windows machine first and test it on your Mac. Otherwise you will have created a cross-platform disk which can't be used across platforms.
On Mac, you can format/reformat a disk as exFAT by opening Disk Utility, selecting the disk on the left (an external connected drive, it shouldn't be your main boot drive!), and clicking the Partition button. Here you have the option to partition the whole disk, or to create multiple partitions (each with different filesystems), and then choose the Format (the last option is Exfat).
Mac OS already supported FAT32 and older versions of FAT. But exFAT has the advantage of supporting files larger than 4 GB, such as large video files. The other Microsoft/PC file system format, NTFS, also supports large files but OSX doesn't support writing to NTFS disks. Thus exFAT is the lowest common denominator for working with large files accessible on both PC and Mac.
If your drive is suddenly disconnected without unmounting in Finder, you will have problems next time you connect the drive. The drive will be in an "unclean" state and require "fsck-ing", which requires scanning the whole disk and can be a long process (taking over an hour). You can check if that's the case by looking for the "fsck_exfat" process in Activity Monitor.