Several issues in that code:
1. You should not use 'external' variables to create a while() loop. Use a while() loop to get one row at a time. Makes the code much, much simpler. What you have now could easily break if the values for $i or $j get modified incorrectly. Not to mention that variables with names like $i and $j provide no context as to what they hold.
2. No reason to call mysql_result() many times instead of just calling a mysql_fetch_ one time.
As to your question it can be implemented many ways. You can create ONE form and have a submit button for each record which will pass the ID of the record to your edit page. Or, you can make a mini form for each record. Which one you go with will be determined by several factors. You also need to decide what 'value' is passed to the edit page (I'm assuming it is the fkey).
Here is one example:
<?php
require('connect.php');
session_start();
$query = "SELECT fkey, projectid, paying, completionmonth, completiondate, personspecify,
projecttitle, categories, tools, shortdescription, date, time
FROM project
WHERE projectstatus='Open'";
$result = mysql_query($query);
$count = 0;
$projectList = '';
while ($row = mysql_fetch_assoc($result))
{
$count++;
$projectList .= "<br>{$count}<br>\n";
$projectList .= "<br>{$row['projectid']}<br>\n";
$projectList .= "<br>{$row['fkey']}<br>\n";
$projectList .= "<br>{$row['paying']}<br>\n";
$projectList .= "<br>{$row['completionmonth']} month {$row['completiondate']} days <br>\n";
$projectList .= "<br>{$row['personspecify']}<br>\n";
$projectList .= "<br>{$row['projecttitle']}<br>\n";
$projectList .= "<br>{$row['categories']}<br>\n";
$projectList .= "<br>{$row['tools']}<br>\n";
$projectList .= "<br>{$row['shortdescription']}<br>\n";
$projectList .= "<br>{$row['date']}<br>\n";
$projectList .= "<br>{$row['time']}<br><br><br><br>\n";
// I want to insert edit button here but dont know how
$projectList .= "<form action='edit.php' method='post'>";
$projectList .= "<input type='hidden' name='fkey' value='{$row['fkey']}'>";
$projectList .= "<button type='submit'>Edit</button>";
ec$projectList .=ho "</form><br>\n";
}
?>
<html>
<head></head>
<body>
<?php echo $projectList; ?>
</body>
</html>