How to create an Instagram account in a desktop browser

UPDATE 06/11/2012: This is now possible “natively” by going to instagram.com/profile_name. Source.

UPDATE 17/04/2020: But clicking any picture opens a login popup instead of details

Yes, if you know your friend’s Instagram username, you can use Webstagram. Search for his/her username and see his/her photo stream. (E.g.: Ariana Grande’s photos.) However, note that if the respective user’s profile is set to private, you can’t see the photos. Source.

Instagram's own help page on Creating an Account & Username says that you need to download the Instagram app on your device in order to register an account.

Mashable's The Beginner's Guide to Instagram also say that "Instagram has always been an almost exclusively mobile platform. Therefore, you must download the iPhone or Android app to your device in order to register an Instagram account."

It's worth noting that Facebook now own Instagram so it's not unlikely that in future you'll be able to login directly using your Facebook account, but as of yet there's no indication that will be the case.

Update

Instagram now allow you to create an Instagram account from a computer:

1. Go to instagram.com.
2. Enter your email address, create a username and password or click Log in with Facebook to sign up with your Facebook account.
3. If you register with an email, click Sign up. If you register with Facebook, you'll be prompted to log into your Facebook account if you're currently logged out.