while ($row = mysql_fetch_array($result)) - how many loops are being performed?

I'm assuming mysql_fetch_array() perfroms a loop, so I'm interested in if using a while() in conjunction with it, if it saves a nested loop.

No. mysql_fetch_array just returns the next row of the result and advances the internal pointer. It doesn't loop. (Internally it may or may not use some loop somewhere, but that's irrelevant.)

while ($row = mysql_fetch_array($result)) {
   ...
}

This does the following:

1. mysql_fetch_array retrieves and returns the next row
2. the row is assigned to $row
3. the expression is evaluated and if it evaluates to true, the contents of the loop are executed
4. the procedure begins anew

$row = mysql_fetch_array($result);
foreach($row as $r) {
    ...
}

This does the following:

1. mysql_fetch_array retrieves and returns the next row
2. the row is assigned to $row
3. foreach loops over the contents of the array and executes the contents of the loop as many times as there are items in the array

In both cases mysql_fetch_array does exactly the same thing. You have only as many loops as you write. Both constructs do not do the same thing though. The second will only act on one row of the result, while the first will loop over all rows.

It depends how many rows are returned in $results, and how many columns there are in $row?

Considering only one row is returned, only one loop will be done in both cases. Though it will check for the loop entering condition twice on each.

For the first one: your program will go through the loop once for every row in the result set returned by the query. You can know in advance how many results there are by using mysql_num_rows().

For the second one: this time you are only using one row of the result set and you are doing something for each of the columns. That's what the foreach language construct does: it goes through the body of the loop for each entry in the array $row. The number of times the program will go through the loop is knowable in advance: it will go through once for every column in the result set (which presumably you know, but if you need to determine it you can use count($row)).