To start, "#amount" is just a string. What you want is the value of the element with an id of amount; "#amount" is the correct selector for that element, but you need to pass it to the jQuery function to actually select it. So:
var amountInput = $("#amount");
That gives you a jQuery object that contains your text input element. Now, you want the value of that input, so you need to call the .val() function on that jQuery object. However, remember that values for text inputs are stored as strings, so you'll need to convert it to a number before you use it as one.
var amount = parseFloat(amountInput.val(), 10);
The entire thing:
var min = 500.00;
var max = 800.00;
var amountInput = $("#amount");
var amount = parseFloat(amountInput.val(), 10);
if (amount < 500.00){
return ("Your payment must be between £500 and £800");
else if (amount > 800.00)
return ("Your payment must be between £500 and £800");
else return false;
Though you're currently calling your code inside a $(document).ready(function() {...}) call - $(function() {...}) is a shorthand - so it will execute immediately after the page "loads" (actually once the DOM has finished being constructed, but before images and such have finished loading). That's probably not going to be much use, since your user won't have time to enter a value.
Instead, bind it to an event, perhaps clicking a button with an id of validate:
$('#validate').on('click', function(e) {
// code above here
});
I'd also suggest taking a read through some introductory jQuery tutorials.