How do I update my browser

Safari is a built in app and cannot be updated independently. If you are running 8.0.2, it is up to date until 8.1 is available, I believe on Monday. However, if the message you are getting is from a specific site , that usually indicates the site is constructed with a specific browser or browsers in mind rather than following standard HTML conventions. Probably not much you can do about it other than complain to the site's webmaster, check the site to see if it has a mobile version or possibly its own app in the app store, or try an alternate browsers from the app store.

If you're able to, please post a screen shot.
Updating Safari isn't possible. It's one of the fantastic iPad built-in apps (it's part of iOS).

Check Settings > General > Software Update, anyway.

Iis it an Apple site saying that? I'm just curious.

LOL ... I have iOS 8.0.2 ....

https://www.icloud.com/activationlock/

No, you cannot access ww.icloud.com via an iOS device. This isn't new to iOS 8. It has always been that way. iCloud i accessed via apps in iOS.

You must be using another browser use Google or Safari Go to your apps on screen and delete Browser that is causing problems. Bsydd uk

Aactually you can access icloud.com - on an iOS device - by going to that main address, then tapping on the URL address bar, pulling down the resulting window and choosing "Request Desktop Site" ... as I've been doing it, and it works just fine ... :-) ...

Many websites, especially older websites, don't recognize the Safari browser on iPads and iPhones. As a result they give the message that you need to update your browser. Ignore the message.