Google's consolidation of many products and services can be quite useful, since all you need is a single account to use any of them, but it can also make it confusing as to what you need to sign up. Although you can use all of the Google services available to you if you want to, you're by no means forced to. If you already have an email address with your preferred provider, for example, you can use that to open a Google account, avoiding Gmail altogether. You can then use that account just for YouTube, or for any other service.
If you have a Yahoo email account, you'll be able to verify your email address by clicking "Verify Now" and then signing into Yahoo using the window that opens, since both Google and Yahoo support the OpenID standard. If you're using a different email provider, Google will send you a verification message, at which point all you need to do is to check your email for that message and click the verification link inside it. If you have a Yahoo email account but you'd rather not use OpenID to verify, click the "Request a Verification Email" link. Occasionally, you may be required to verify the account using a phone number instead.
Once you've completed the verification, whatever form it took, you'll be able to sign into YouTube using your current email address and the password you chose when you registered for the Google account.
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