You can use the contenteditable attribute on the cells, rows, or table in question.
Updated for IE8 compatibility
<table>
<tr><td><div contenteditable>I'm editable</div></td><td><div contenteditable>I'm also editable</div></td></tr>
<tr><td>I'm not editable</td></tr>
</table>
Just note that if you make the table editable, in Mozilla at least, you can delete rows, etc.
You'd also need to check whether your target audience's browsers supported this attribute.
As far as listening for the changes (so you can send to the server), see contenteditable change events
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