The basic idea is simple, and obvious in hindsight. With Follow-Up Mode, Alexa will listen for 5 seconds after she's done responding to you to see if you have anything else to ask. The blue ring on your Alexa device of choice will light up during these 5 seconds to indicate that she's listening. If you have anything else to ask her, you can just ask it without needing to say her name first. If you don't, she'll just go back to sleep automatically without saying anything else.
Amazon says Alexa will only respond to a follow-up commands if she's 'confident' that it is a legitimate request and not just background noise. To end a 'conversation' you can say something like 'thank you' or 'stop' while Alexa is listening for a follow-up.
You can enable Follow-Up Mode on your device by opening the Alexa app and navigating to the Settings menu. Select your device then toggle the Follow-Up Mode switch.
The feature should be available to the entire Echo lineup and some third-party Alexa devices; however, it's only available for US English and won't activate while you're listening to music, streaming an audiobook, or making a call.
Amazon's Alexa devices include the $49.99 Echo Dot, to the $99.99 Echo, and the $229.99 Echo Show. Check out the full range here.