What has caused this error message: mysqli_real_escape_string() expects 2 parameters.

you need to pass the connection resource/object as the second parameter -ex:
$conn = mysql_connect("localhost","username","password");
...
$first_name = mysql_real_escape_string(trim($_POST['first_name']), $conn);

Is there any difference between mysqli_real_escape_string and mysql_real_escape_string?

Also, why did my INSERT statement work (I have that posted below)? I used the same format but didn't get an error?

$cxn = mysqli_connect($host,$user,$password,$database)
or die ("couldn't connect to server");
 
$first_name = mysqli_real_escape_string(trim($_POST['first_name']));
 
$last_name = mysqli_real_escape_string(trim($_POST['last_name']));
 
 
$text = $_POST['bio'];
$textBr = nl2br($text);
$finaltext = mysqli_real_escape_string($textBr);
 
$city = mysqli_real_escape_string($_POST[city]);
 
$insert = "insert into winners (first_name, last_name, email, state, bio, radio_id, image_file, mp3_file, city, press_release, region, winner_type)
values ('$first_name','$last_name', '$_POST[email]', '$_POST[State]', '$finaltext', '$_POST[radio_id]', '$_POST[image_file]', '$_POST[mp3_file]', '$city', '$_POST[press_release]', '$_POST[region]','$_POST[winner_type]')";
$insertexe = mysqli_query($cxn, $insert)
or die ("Couldn't execute query.");

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>>Is there any difference between mysqli_real_escape_string and mysql_real_escape_string?
They are meant to serve the same purpose.

If you are interested in the differences, refer to:
http://www.johnjawed.com/benchmarks/

>>why did my INSERT statement work
Because the syntax is perfectly valid. What you got is a "Warning" not an "Error". If you get a "Warning" you get the annoying messages you got, but execution of the script does NOT stop.

$cxn = mysqli_connect($host,$user,$password,$database)
or die ("couldn't connect to server");
 
$first_name = mysqli_real_escape_string(trim($_POST['first_name']), $cxn);
 
$last_name = mysqli_real_escape_string(trim($_POST['last_name']), $cxn);
...

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sorry, the connection resource goes first in mysqli_real_escape_string:

$cxn = mysqli_connect($host,$user,$password,$database)
or die ("couldn't connect to server");
 
$first_name = mysqli_real_escape_string($cxn, trim($_POST['first_name']));
 
$last_name = mysqli_real_escape_string($cxn, trim($_POST['last_name']));
...

OK, it's as clear as mud right now and it ain't for lack of trying...

hielo, it appear as though you made a change between your first and second posts. You begin your second post by saying that the connection resource needs to be positioned first. Yet, when I look at your code, I don't see any difference. It seems as though the connection resource is first in both instances.

NerdsOfTech - Is there no way to write an UPDATE statement using mysqli? I want to learn WHY something works and not just WHAT and that's one of the reasons I appreciate both you ninjas weighing in on this stuff because you offer not just a solution, but some commentary as well. My thing is, I want to use mysqli. Can you show me how to make it work in that context?

It's my browser. I didn't notice until you pointed it out that there was more to the code that I initally saw in the "box" where you published your suggestion. I got it now...

glad to help