The "save password" part comes from the browser's password manager whenever it sees an <input type="password">
that looks like it really is asking for a password. You can use the autocomplete attribute to suppress this in most browsers:
<input type="password" name="password" autocomplete="off">
This won't validate but that usually doesn't matter.
The "remember me" part is completely separate from the browser's password manager. The "remember me" flag is the server's business and all it does is fiddle with the expiry date on the cookie that it sends back. The server will always send a cookie back (unless they're not using cookies for tracking sessions but that's rare and wouldn't need a "remember me" anyway) with something inside it to identify the client user.
If you check "remember me" then you're telling the server that you want a persistent session. To achieve this, the server will include an expiry date with the cookie and that expiry date will be some time in the future. When the date arrives, the browser will expire and delete the cookie; without the cookie, the server won't know who you are anymore and you'll have to login again.
If you don't check "remember me" then you'll get a session cookie. Session cookies don't have expiry dates on them so automatically expire when the browser exits. Session cookies are useful for shared machines.
Executive summary:
- "Save password" is from the browser's password manager.
- "Remember me" is about the login cookie's expiry time.
Sorry to be so long winded but there seems to be some confusion and a lack of clarity in the other answers.
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