If you're disloyal, you're not faithful or trustworthy — you can't be depended on by those who put their trust in you. It would be disloyal to join in with a group of people gossiping about your best friend.
It's disloyal when a country violates a signed treaty, and it's disloyal for a citizen to spy for a foreign state against its own government. But it's also disloyal to betray the trust of someone in your life. A disloyal sibling tattles on his brothers and sisters, and a disloyal friend won't keep your secrets. This adjective comes from the Old French desloial, "treacherous or deceitful," which adds the prefix des-, "the opposite of," to loial, "faithful."