There are four different methods to back up (or not) your iOS device, let's take a look at what each flavor of backup does. Android users may have more/fewer options; I don't know, but I really encourage you to find out, or if someone wants to give me an Android device I can look into those options as I have time. (I hear the Nexus 7 is nice, and it fits nicely into my Android buying guide)
In ye olden days of cell phones, if you broke your device everything was gone and you were sad. If you had a GSM phone with a SIM card, perhaps some or all of your address book could be stored there if you 1, knew about the option and 2, bothered to do it (not just once!). Not backing up your iPhone at all is a callback to these heady days where every trip to the restroom or walk under cloudy skies was skirting danger. Except that losing everything will drive you up the damn wall because you put months or years worth of data, logins, media, etc. into that phone because they're computers now, and now all of that is gone. The only consolation is that you won't have to re-purchase your apps or media, but that is a cold comfort when composing your second or third "I lost my phone, please send me your #" status update on Facebook.
If you sync your iPhone to your Mac or PC iTunes can backup your device as part of the process. Since iOS 5 you can also sync wirelessly, so you don't even have to take it out of your pocket if you don't want. When your device is connected, click on it in iTunes' Devices section. On the Summary tab, make sure "Back up to this computer" is selected. Just remember to sync now and then (it's a fine way to charge, you know) and you're set.
There is a caveat though: by default the "Keychain" (where your email account, website, and most app passwords are stored) cannot be restored to a new device. This means that if you drop your phone into a bucket of water, when you bring home your new phone, all of the media, apps, etc. will be put back just as they were, but you will have to re-enter every password. To enable restoration of your Keychain, you need to select the "Encrypt local backup" option and supply a password. Don't forget your password, or you'll lose the entire backup rather than just your Keychain.
Since iOS 5 was released, it is becoming less and less necessary to ever connect an iOS device to a computer to manage it, including backup. Once a day your device is just backed up to Apple's iCloud servers automatically (though you can also trigger backups on demand). There are some rules though, the device has to be: 1. plugged in and charging, 2. locked, and 3. connected to a wireless network (3G will not do, as it would rapidly wreck your data limit). This can be turned on either by connecting the device to a computer and selecting the option in iTunes, or by going into the iCloud option in the Settings app.
You can tailor what is and isn't backed up to iCloud in the Settings app, and you'll likely want to keep an eye on that because you only have 5GB of space unless you purchase some. If you take a lot of photos, that will disappear faster than you think, though only using PhotoStream and not backing up your photos will (almost!) be as good. It is also important to note though that the list of items backed up to iCloud is much smaller than the list of items backed up by iTunes, and there is no way to restore a Keychain to a new device. Bear in mind that if you do have convenient access to a laptop or desktop that may be a better option. Or perhaps…
You can have your Keychain and eat it too, something like that anyway. Selecting the iCloud backup option disables automatic iTunes backups, but it doesn't stop you from doing them yourself. All you have to do is plug in your device each night to automatically backup to iCloud, and then when you want to do a full backup, connect the device to your computer, let it sync, and then right-click on it in iTunes' Devices section. There will be a Back Up option that you can choose to do a regular iTunes backup, even if you have iCloud backups enabled. It's only really necessary to backup to iTunes after telling a new app to remember your password, or changing an existing password in an app. Should disaster strike, iTunes can give you back your most recent full backup and your Keychain, and iCloud can fill in the gaps.
I wrote this for two reasons: 1, Everybody should be backing up their computers, and modern smartphones are just as much computers as the ones on your desk or lap; and 2, I have had occasion to test this recently. I frequently hook my iPhone up to my Mac, so I just use the iTunes backup (it's inconvenient for me to charge it overnight) and that's fine. This is also the fastest and easiest way to get new episodes of podcasts on the phone until the Podcasts app works properly.
A few weeks ago though, I was going to be in northern Indianapolis and for some time I'd had some trouble with my Home button working properly (that's kind of a big deal on an iPhone). I just thought I'd go see if the Apple Store had any ideas or a way to clean it, but about a minute after explaining my problem a newly refurbished phone was staring me in the face. I wasn't expecting to replace my phone that day, but I had backed my phone up the night before and that home button was really bugging me. Shortly after getting home I hooked up the new phone and restored my latest backup to it. Short of setting up the notification options for a few apps (push notifications are weird for boring and off-topic reasons), it was as if my phone never had a problem. It was easy and I didn't have to ask for phone numbers on Facebook with my hat in my hands.
So, if you own an iPhone (or iPod Touch, iPad, etc.) check on your backup settings, you don't want to be learning about this stuff after your device is busted. For Android users I'm sure there's something similar, just look around, but be wary of anything that comes from the device manufacturer; the backup that Samsung software makes probably won't help you if you move to an HTC device later.