Ok, you've got a few problems there..
Firstly, please avoid opening php tags and closing them constantly. Just echo back the HTML. Going in and out of php so often will cause performance issues.
So, first off, you need to set up a form on here, your form needs to point to a second page (or self-process using an isset function, but let's go with the former solution for clarity). The second page is what's going to dump things into the DB for you based on what the user ticked.
So, here's the HTML for your form;
<form action="database.php" method="POST">
<div>
<label for="checkbox1">This is the first option:</label>
<input type="checkbox" id="checkbox1" name="checkbox1" />
</div>
<div>
<label for="checkbox2">This is the second option:</label>
<input type="checkbox" id="checkbox2" name="checkbox2" />
</div>
<div>
<input type="submit" />
</div>
</form>
Again, I'd recommend echo'ing this back in your foreach loop - a short function could accomplish this for you.
So, once someone hits the "Submit" button, they'll be taken to your second page (database.php as we've called it), and their options will be stored in the POST array. For good practice, always use POST rather than GET when dealing with DB entries.
So on this page, we'd check what checkboxes were selected, and then update the db as required;
if (isset($_POST['checkbox1']))
{
// user ticked checkbox1
$sql = "UPDATE table
SET 'checkbox1' = 1
WHERE 'user' = 'username';"
mysqli_query($sql);
}
Note that's just a rough solution - you haven't told me exactly how you're doing this, so I'm assuming your DB is tied to user's votes etc. You can also update checkbox1 by incrementing the value.. etc etc.
Hope this helps.