The button works in much the same way as a long press on the home button does currently on most Android devices. A single tap of the button will launch the Google Assistant and get it to listen to what you want it to do. A double tap essentially shows you Google Now-style information - a curated view with personalised and helpful information based on the time of day, location and your recent interactions with the Assistant.
Long-pressing the button activates a walkie talkie feature, so you can make sure the Assistant will listen to your full query - this is most useful for dictating emails or long text messages.
Naturally, the Google Assistant can perform various actions such as getting answers to questions, viewing your calendar, finding directions to the nearest coffee shop, starting music/podcasts, and control smart home devices with your voice.