Ceejayoz & Joe have already covered the 2 basic options, Javascript or IFrame.
If you want to check out some examples, take a look at the StackOverflow flair feature, that does exactly what you are trying to do and allows a user to embed the StackOverflow user info box on their own website.
If you want to see how it's implemented, check out the html which a user would link to with an IFrame, or this Javascript, which the user would call from their page. (see the notes under "how do I use it" on the SO flair page.)
[You need to insert your user number in those links to see the relevant html/js files for yourself.]
In the case of SO, the js/html is generated specifically for each user. For completeness I've included the StackOverflow js/html that achieves this.
JavaScript: You can see when executed this basically writes a script element into the users html page that injects a link to the SO css file into the head. Then it just includes some fairly simple div/span tags which get styled by the css. Obviously, the user could in this case provide their own css instead if they wanted to style it differently.
document.write('
<script>
var link = document.createElement(\"link\");
link.href = \"http://stackoverflow.com/content/flair-Default.StackOverflow.css\";
link.rel = \"stylesheet\";
link.type = \"text/css\";
var head = document.getElementsByTagName(\"head\")[0];
head.appendChild(link);
</script>
<div class=\"valuable-flair\">
<div class=\"gravatar\">
<a title=\"See my profile on Stack Overflow\" target=\"_blank\" href=\"http://stackoverflow.com/users/1/jeff-atwood\">
<img src=\"http://www.gravatar.com/avatar/51d623f33f8b83095db84ff35e15dbe8?s=50&amp;d=identicon&amp;r=PG\" height=\"50\" width=\"50\" alt=\"\">
</a>
</div>
<div class=\"userInfo\">
<span class=\"username\">
<img src=\"http://sstatic.net/so/favicon.ico\" />
<a href=\"http://stackoverflow.com/users/1/jeff-atwood\" target=\"_blank\">Jeff Atwood</a>
<span class=\"mod-flair\" title=\"moderator\">♦</span>
</span>
<br />
<span class=\"reputation-score\" title=\"reputation score\">16,907</span>
<br />
<span title=\"5 gold badges\"><span class=\"badge1\">●</span>
<span class=\"badgecount\">5</span></span><span title=\"55 silver badges\">
<span class=\"badge2\">●</span><span class=\"badgecount\">55</span></span>
<span title=\"72 bronze badges\"><span class=\"badge3\">●</span>
<span class=\"badgecount\">72</span></span>\r\n </div>\r\n</div>'
);
Html - for an IFrame: You can see this is a full HTML document (XHTML actually to be pedantic) that includes everything needed including the link to the css in the head and the content in the body.
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link rel="stylesheet" href="http://sstatic.net/so/flair-Default.StackOverflow.css" />
</head>
<body>
<div class="valuable-flair">
<div class="gravatar">
<a title="See my profile on Stack Overflow" target="_blank" href="http://stackoverflow.com/users/1/jeff-atwood"><img src="http://www.gravatar.com/avatar/51d623f33f8b83095db84ff35e15dbe8?s=50&amp;d=identicon&amp;r=PG" height="50" width="50" alt=""></a>
</div>
<div class="userInfo">
<span class="username"><img src="http://sstatic.net/so/favicon.ico" /><a href="http://stackoverflow.com/users/1/jeff-atwood" target="_blank">Jeff Atwood</a><span class="mod-flair" title="moderator">♦</span></span>
<br />
<span class="reputation-score" title="reputation score">16,907</span>
<br />
<span title="5 gold badges"><span class="badge1">●</span><span class="badgecount">5</span></span><span title="55 silver badges"><span class="badge2">●</span><span class="badgecount">55</span></span><span title="72 bronze badges"><span class="badge3">●</span><span class="badgecount">72</span></span>
</div>
</div>
</body>
</html>