For a given website / web-application, you can use react either client-side, server-side or both.

Client-Side

Over here, you are completely running ReactJS on the browser. This is the simplest setup and includes most examples (including the ones on http://reactjs.org). The initial HTML rendered by the server is a placeholder and the entire UI is rendered in the browser once all your scripts load.

Server-Side

Think of ReactJS as a server-side templating engine here (like jade, handlebars, etc...). The HTML rendered by the server contains the UI as it should be and you do not wait for any scripts to load. Your page can be indexed by a search engine (if one does not execute any javascript).

Since the UI is rendered on the server, none of your event handlers would work and there's no interactivity (you have a static page).

Both

Here, the initial render is on the server. Hence, the HTML received by the browser has the UI as it should be. Once the scripts are loaded, the virtual DOM is re-rendered once again to set up your components' event handlers.

Over here, you need to make sure that you re-render the exact same virtual DOM (root ReactJS component) with the same props that you used to render on the server. Otherwise, ReactJS will complain that the server-side and client-side virtual DOMs don't match.

Since ReactJS diffs the virtual DOMs between re-renders, the real DOM is not mutated. Only the event handlers are bound to the real DOM elements.