With Account Chooser, you can personalize the Google sign-in page to remember the accounts you access frequently. You can conveniently select an account and enter its password to sign in.
If you want to add an account to Account Chooser, simply sign in and leave the "Stay signed in" checkbox selected. If you want to remember more accounts, just click "Add account" in Account Chooser to take you to the sign-in page and then leave the "Stay signed in" checkbox selected. If you want to remove an account from Account Chooser, sign out, then go to a Google sign-in page and click the Remove button below the account list.
The main takeaway is to uncheck "stay signed in" if you're using a public computer or a friend's computer. Google suggests something different, but I think it's a bad idea. "If you're signing in to your Google Account from a device that only you use (for example, a personal laptop), we recommend leaving the 'Stay signed in' checkbox selected. This ensures that you won't be constantly interrupted to re-enter your password and makes your future sign-ins easier with Account Chooser. For shared devices (a computer at an Internet cafe, for example), we still recommend that you leave the checkbox selected."
There's also a help center article about Account Chooser and federated sign-in:
"To make it easier for you to sign in to your accounts in the future, Google's sign-in page will, by default, display a list of accounts that you've previously used with a specific browser. To sign in, simply click the account entry (which includes your name, e-mail address, and photo if you have one) for the account you'd like to use. You'll then be redirected to the identity provider for your account (Yahoo mail, Hotmail, AOL mail, etc.). If you haven't typed your password recently, your identity provider may ask you to type it again. If you'd rather not use an identity provider for your Google Account, you can enable 2-step verification, or change the email address on your account to a Gmail address."
The page includes two useful links for enabling Account Chooser and disabling Account Chooser.
"If you're using a shared computer and you don't want other users to notice that you've used this computer with a particular account, you should consider using the incognito or private browsing feature of popular browsers. This way, your account entry won't appear listed on the sign-in page when another user goes to sign in," suggests Google.
This OpenID site has more information about account choosers: "Websites are replacing traditional login boxes with an account chooser to make those sites more secure and easier to use. When you try to sign in to such a site, you will see a page with the list of accounts you most frequently use on this computer."
Google tested the account chooser last year and now it looks like this feature will be enabled by default. A blog post from 2011 mentioned that "the so-called Account Chooser user interface is a replacement for traditional email/password log-in boxes used on many Web sites. The spec is part of an on-going campaign by Google and others to eliminate passwords in favor of user log-ins that rely on secure tokens which pass only authentication verification."