The browser sends a HTTP (http://www.w3.org/Protocols/rfc2616/rfc2616.html) request, which includes the URL, the request method (GET, POST etc), cookies and a whole bunch of other stuff. Here's the request from my browser to this SO page:
GET /questions/2575970/how-are-cookies-sent-to-a-website HTTP/1.1
Host: stackoverflow.com
User-Agent: Mozilla/5.0 (Windows; U; Windows NT 6.1; en-GB; rv:1.9.2.3) Gecko/20100401 Firefox/3.6.3
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,*/*;q=0.8
Accept-Language: en-gb,en;q=0.5
Accept-Encoding: gzip,deflate
Accept-Charset: ISO-8859-1,utf-8;q=0.7,*;q=0.7
Keep-Alive: 115
Connection: keep-alive
Cookie: __utma=140021253.1463780230058740000.12348924611.1279210754.1270438283.1398; __utmz=140222553.12686423964.1149.21...
If-Modified-Since: Sun, 04 Apr 2010 21:30:58 GMT
Note that the cookie doesn't normally contain the user name, just a index to a lookup table that's stored server-side.