I don't know of any System.Net.WebBrowser, but WebClient is basically a class that lets you easily download files (including html pages) from the web into memory or even directly to file.  A basic code sample looks like this:
string html;
using (var wc = new WebClient())
{
    html = wc.DownloadString("http://stackoverflow.com/questions/1780679/");
}
You can do a lot with WebClient, but there are some limitations.  If you need to do some serious web scraping, you'll need to get lower level.  That's where the HttpWebRequest/HttpWebResponse come in.  You can use them to send any request a normal web browser might send, in any sequence.  For example, you may need to authenticate with a web site before you can request the page you really want, and WebClient might not be able to do that.  HttpWebRequest will.
Now, there is one other option.  System.Windows.Forms.WebBrowser is a control designed to place on a form.  It basically wraps the engine used in Internet Explorer to provide all the capabilities of a web browser.  You need to be careful using this for general scraping:  it's not portable (bad for mono), uses a lot of resources, has similar security issues as running a full browser, and has side-effects such as potentially leaking popup windows.  The control is best used in a form to connect to a specific known web resource. For example, you may have a Windows Forms app for sale, and web app where you sell it for download.  You might provide a WebBrowser control that shows a few pages on this web site specifically intended for view in your app that allows users to purchase in-app upgrades.