Copyvar arr = [3, 6, 2, 56, 32, 5, 89, 32];
var largest = arr[0];

for (var i = 0; i < arr.length; i++) {
  if (arr[i] > largest ) {
    largest = arr[i];
  }
}
console.log(largest);

• You need to define i or else it become a global variable.
• Don't redefine largest in the loop.
• Since you're looping through the array, use i < array.length instead of i <= largest.
• Since you're comparing each of the items in the array to largest, use if(largest < array[i]) instead of if(array > largest)
• You should set largest equal to the first element in the array because what if all the numbers are negative?
• array is a bad variable name because it's too similar to Array (the array constructor). Try arr instead.

One liner:

var largest = Math.max.apply(0, array);

More info here: Javascript max() function for 3 numbers

How about augmenting the built-in Array object to use Math.max/Math.min instead:

Array.prototype.max = function() {
  return Math.max.apply(null, this);
};

Array.prototype.min = function() {
  return Math.min.apply(null, this);
};

let p = [35,2,65,7,8,9,12,121,33,99];

console.log(`Max value is: ${p.max()}` +
  `\nMin value is: ${p.min()}`);

Here is a JSFiddle.

Augmenting the built-ins can cause collisions with other libraries (some see), so you may be more comfortable with just apply'ing Math.xxx() to your array directly:

var min = Math.min.apply(null, arr),
    max = Math.max.apply(null, arr);

Alternately, assuming your browser supports ECMAScript 6, you can use spread syntax which functions similarly to the apply method:

var min = Math.min( ...arr ),
    max = Math.max( ...arr );