Since, The web client acts as a mirror for your WhatsApp mobile, you'll need to have both, PC and Mobile connected to an active internet connection, in order to use it. WhatsApp has pushed updates for its mobile apps, so you'll need to update your app in the first place and then you'll be able to see "WhatsApp Web" option available in the Menu. Once you've your WhatsApp updated in the mobile, head over to WhatsApp Web, you'll see, a QR code on screen that you'll have to scan with your WhatsApp mobile device in order to log in. Unfortunately, WhatsApp's web client is not available for iOS users due to Apple's platform limitations.
WhatsApp Web works with Google Chrome only, so if you prefer Opera, Firefox, Safari, IE or any other browser you're not in luck for now, you'll either need to download chrome or you could wait for its compatibility with other browsers. Although, the web version is just like the WhatsApp mobile app, but there are certain things that can't be done on the web client, for example, you can't block someone from WhatsApp using its web interface, you can't create or remove groups over the web, and you can't change your Status or Profile Picture. WhatsApp web has some limitations for now, but in the future, it's going to be a big hit.