Alexa's personality styles let you change the way Alexa answers without changing what it's able to do. Amazon currently offers five personality options: Brief, Chill, Sweet, Sassy and Original. Brief is exactly what it sounds like, for those who prefer quick, concise answers. Chill is casual and conversational. Sweet is warmer and more encouraging, and Sassy adds a more playful, witty edge. The latter comes with an "adults only" warning that you have to tap to proceed. It's not exactly NSFW or anti-kid-friendly, but Alexa may make fun of your question and use a few colorful pejoratives.
The quickest method is to say, "Alexa, change your personality style." Alexa will present your options and guide you through them. If you prefer to use the app, open Devices > Your Device > Device Settings > Alexa's Personality Style. From there, simply swipe through the available personalities, left to right, and you can return to the original style at any time.
Again, personality styles are device-specific, so if you want a consistent personality throughout every Echo device in your home, you'll have to change each one individually.