Semantic HTML means using HTML tags for their implied meaning, rather than just using (meaningless) div and span tags for absolutely everything. Why would you want to do this? Depending on the tag, the content in the tag can be interpreted in a certain way. Here are some examples.
If you use <h1> instead of <div class="header">, and <h2> instead of <div class="subheader">, et cetera, Google and other search engines will interpret your headers as being important titles in your page. This way, when people search on the words in your headers and sub-headers, your page will be considered more relevant (and rank higher). Plus, it's much shorter and cleaner.
This works both ways: don't use header tags for anything except headers, especially not increasing your font size or outlining your search engine keywords. This way, your page can be parsed for structure (you can do this with the W3C HTML Validator). This structure can then be used by screen readers or other tools to build a table of contents for your page.
The <label> tag is so sadly forgotten. It's not immediately clear what the point of using it is, so very few web pages take advantage of it. The label tag is used to identify a label for an input field, for example "E-mail Address". It can either be used be wrapping it around the text and input field like:
<label>First Name: <input name="fname" />
or it can be used with the for attribute like so:
<label for="fname">First Name: <input id="fname" name="fname" />
Why use the label tag instead of <div class="label"> ? Well, it's shorter and cleaner. But it also let's screen readers and other tools identify the text associated with an input field. Without using the label tag, it can be very difficult for some people to know what is supposed to go into your form fields.
These days, everyone's moving away from using tables. This is great because tables aren't intended for structuring the way your web page looks. But tables still have a very important purpose. Any time you need to display data that would go in a spreadsheet, tables are here to help.
When using tables, there are a number of tags and attributes that aren't widely used, but are very important for accessibility. Use the summary attribute to give a longer summary of the data in the table. Use the <caption> tag to give a brief title to the data. Use <th> tags to identify the column and row headers in your table. Then, you may want to use the headers attribute on the <td> tags to identify which headers apply to that cell. For more examples and details on accessibility with tables, see the W3C's Accessibility Guidelines.
Lists are the new tables. Whereas tables are intended for grids of data, lists are intended for lists of content. This is great for us, because most web pages are essentially lists of different things. For example, look at this site. On the front page, I have a list of blog entries in the centre. On the sides, I have lists of links (archive, categories, et cetera), and the sides themselves are lists of lists. If I had used tables, I would've been saying "this stuff on the left has something to do with the stuff in the middle", but it doesn't, really. By using lists, I'm simply saying "this stuff is a list of items that have something to do with each other", which they do.
You have three types of lists to choose from, but choose wisely. There are Ordered Lists (<ol>), Unordered Lists (<ul>), and Definition Lists (<dl>). Only use Ordered Lists when the entries have some kind of order. Use Definition Lists any time you need name/value pairs, or when you need to break your list up into sections. The rest of the time, Unordered Lists are a safe bet.
Lists not only give structure to your page, they're incredible handy for styling. You can just put an id or class on the outer tag (eg. <ul>), then style both the outer tag, and the inner <li> tags.
Try to use the full variety of HTML tags whenever possible. Sometimes you'll be stuck with using < div > tags, but try to limit them to whenever you can't find a suitable HTML equivalent. At the same time, try to avoid using HTML tags for anything except their intended purpose. By doing this, your HTML will be cleaner, and its structure will be more readable and understandable - not just to people but to screen readers, search engines, and other programs and tools.