In 2013 Google released a major re-design of the Google sign-in process changing both its look and function. In 2017 Google released a minor update with some small changes and also made the look-and-feel consistent across all platforms (computers and mobile devices). This article will walk through how the sign-in page works now and the options available.
Typically when signing into Google you will be signing into a specific product, like Gmail. Many products have their own initial landing page. For Gmail, when you go to https://mail.google.com/ it will look something like this.
This page, or one like it, is what you may see the first time you try to sign into some Google product on a computer. This is basically an advertisement for that product and since it's your first visit it emphasizes the buttons to create a new account. If you already have an account, click the "SIGN IN" link in the upper/right. If this is the first time signing in on this computer or device, you'll see the following pages to enter your e-mail address and password to sign into your account.
When you are done with your account, you should sign out. Be aware that just closing the browser does not also sign you out of your account.
This returns you to the above page asking for your password to sign into the account again. If you want to sign into a different account, click the small 'v' symbol on the right of the password entry page to go to a page where you can request to "Use another account".
Just as above, you would enter the new e-mail address and password to sign into a different account. But now when you sign out you would see the page above with multiple accounts listed.
If you select a listed account, you will see the password screen where you can finish logging in. Or you can add another account to the list like before. If you wish to remove some or all of the listed accounts, click on "Remove an account".
Clicking the 'X' to the right will remove that account. Click "Done" at the bottom when finished.
There are some additional options displayed as blue text on the Sign In and Password pages seen above. "Forgot email?" will start the process to request a forgotten e-mail address from Google. Likewise, "Forgot password?" will start the process of lost password recover. Both of these are documented in detail in the following article: http://gmailaccountrecovery.blogspot.com/
The "More options" link will display a small context-dependent menu which includes "Create account" and perhaps some other platform dependent choices.
Hopefully that makes the changes to the sign-in process a bit more clear Just remember that you are signing into your Google account, in order to access a specific product. Once you have signed into your Google account you have access to all the other products and services you use in that account.