If you choose to set up your iPhone as a new device--or after you choose an iCloud backup to restore from--the first thing you have to do is supply an Apple ID, or create one if you don't have one. If you've ever purchased something from the iTunes Store, you've signed up for an Apple ID (it's usually your primary email address). Your login information for Apple's iCloud service should also work for signing in.
Use your current Apple ID: Already have an Apple ID? Tap the Sign In With An Apple ID button and enter your username (usually your email address) and password. Apple then spends a few moments linking your device to your Apple ID.
Sign up for a new Apple ID: If you don't have an Apple ID, it's easy enough to create one by tapping the Create A Free Apple ID button. You need to enter your birthday, name, email address (or create a new iCloud email address), a password, a security question (in case you forget your password), and whether you'd like to receive email updates from Apple. Once you've entered all your information, you're asked to read and agree to the terms and conditions, and Apple then registers your Apple ID.
No Apple ID for me: If you'd rather not set up an Apple ID, you can tap the Skip This Step link in the lower right corner. You can always add or create one from the Settings app later, but note that you won't be able to buy anything from the iTunes Store or set up iCloud until you do.
Once you've set up an Apple ID, you can also set up iCloud on your device. iCloud is an umbrella term for Apple's collection of syncing services, and it allows you to sync your photos, apps, contacts, calendars, reminders, notes, and mail across multiple devices.
Choose to set up iCloud, and you're first asked whether you'd like to enable iCloud backups for your iPhone. If you do so, you can have your device back up all essential settings to your iCloud account; if you ever need to restore, you can do so over Wi-Fi without an additional computer. You can also elect to use iTunes to back up your iPhone to your computer.
Additionally, you're asked whether you'd like to opt in to iCloud's Find My iPhone service. This enables location monitoring for your device, allowing you to find it using your Apple ID and the Find My iPhone app, should it go missing.
One of the iPhone's nice features is Apple's personal voice assistant, Siri. If you want to take advantage of Siri, you can enable it here. Then, once you finish the setup process, you're asked if you'd like to send Apple anonymous diagnostics and usage information (similar to a desktop crash report). After you answer that question, your iPhone will be all set and ready for you to begin using.