Scandalous describes something that's shocking, and maybe a little embarrassing or even offensive. Your great-grandfather might be a scandalous character in your family history if he was married seven times, each time to younger and younger wives.
The adjective scandalous can refer to something morally offensive, or even illegal, although it's used often simply to mean "shocking". It describes something that has potential to cause a scandal, which comes from the Greek skandalon, "a trap laid for an enemy." This idea of a trap or a snare is a great metaphor; you can stumble into scandal as if it's just waiting for you to make a misstep, and if you do the results will be scandalous.