Which file format should I give it? I need to backup all my data, perhaps FAT32? what is ExFAT? and all those other file formats? which is best?
You haven't said what you're backing up, but I'm assuming it's the internal disk of a Mac, or perhaps an external drive connected to a Mac. In that case, you almost surely want to format as "Mac OS Extended (Journaled)". That's the default for Macs (at least relatively recent Macs -- my experience only goes back to 2008).
Picking one of the "Mac OS Extended" flavors will ensure that all of the extended attributes for your Mac files will be able to be copied to the new disk, and you probably don't need the Case-Sensitive or Encrypted flavors unless you have special requirements or desires. Thus, "Mac OS Extended (Journaled)".
However, if you need to be able to read your new disk from a Windows machine, you might use the less desireable FAT or exFAT. I think that's the only common reason to use them. FAT is widely supported, but individual files are limited to 4 GB and the volume size to 2 TB. exFAT raises those limits, but still doesn't support OS X extended attributes or ACLs, to my knowledge.
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What's the proper way to give file format to a brand new external hard drive?
It's a Seagate Expansion 1TB Portable Drive (USB 2.0, 3.0)
I know it involves opening the Disk Utility app, selecting the drive, and then going into Erase? what else?
This is what I do:
open Disk Utility
select the new drive
pick the "Partition" tab
Partition Layout: 1 partition
give it a name
format: Mac OS Extended (Journaled)
size: leave it alone
options: verify that it defaults to "GUID Partition Table" (the newest format)
click Apply
If you do that, you should be fine. (WARNING: this erases ALL files on the disk you've selected in Disk Utility! But I hope you knew that... if it's a brand-new drive this is OK.)