For 2 dimenional Arrays:

for(var i = 0; i < parentArray.length; i++){
    for(var j = 0; j < parentArray[i].length; j++){

        console.log(parentArray[i][j]);
    }
}

For arrays with an unknown number of dimensions you have to use recursion:

function printArray(arr){
    for(var i = 0; i < arr.length; i++){
        if(arr[i] instanceof Array){
            printArray(arr[i]);
        }else{
            console.log(arr[i]);
        }
    }
}

Yes, assuming your implementation includes the for...of feature introduced in ECMAScript 2015 (the "Harmony" release)... which is a pretty safe assumption these days.

It works like this:

// REQUIRES ECMASCRIPT 2015+
var s, myStringArray = ["Hello", "World"];
for (s of myStringArray) {
  // ... do something with s ...
}

Or better yet, since ECMAScript 2015 also provides block-scoped variables:

// REQUIRES ECMASCRIPT 2015+
const myStringArray = ["Hello", "World"];
for (const s of myStringArray) {
  // ... do something with s ...
}
// s is no longer defined here

(The variable s is different on each iteration, but can still be declared const inside the loop body as long as it isn't modified there.)

A note on sparse arrays: an array in JavaScript may not actually store as many items as reported by its length; that number is simply one greater than the highest index at which a value is stored. If the array holds fewer elements than indicated by its length, its said to be sparse. For example, it's perfectly legitimate to have an array with items only at indexes 3, 12, and 247; the length of such an array is 248, though it is only actually storing 3 values. If you try to access an item at any other index, the array will appear to have the undefined value there, but the array is nonetheless is distinct from one that actually has undefined values stored. You can see this difference in a number of ways, for example in the way the Node REPL displays arrays:

> a              // array with only one item, at index 12
[ <12 empty items>, 1 ]
> a[0]           // appears to have undefined at index 0
undefined
> a[0]=undefined // but if we put an actual undefined there
undefined
> a              // it now looks like this
[ undefined, <11 empty items>, 1 ]

So when you want to "loop through" an array, you have a question to answer: do you want to loop over the full range indicated by its length and process undefineds for any missing elements, or do you only want to process the elements actually present? There are plenty of applications for both approaches; it just depends on what you're using the array for.

If you iterate over an array with for..of, the body of the loop is executed length times, and the loop control variable is set to undefined for any items not actually present in the array. Depending on the details of your "do something with" code, that behavior may be what you want, but if not, you should use a different approach.

Of course, some developers have no choice but to use a different approach anyway, because for whatever reason they're targeting a version of JavaScript that doesn't yet support for...of.

As long as your JavaScript implementation is compliant with the previous edition of the ECMAScript specification (which rules out, for example, versions of Internet Explorer before 9), then you can use the Array#forEach iterator method instead of a loop. In that case, you pass a function to be called on each item in the array:

var myStringArray = [ "Hello", "World" ];
myStringArray.forEach( function(s) { 
     // ... do something with s ...
} );

You can of course use an arrow function if your implementation supports ES6+:

myStringArray.forEach( s => { 
     // ... do something with s ...
} );

Unlike for...of, .forEach only calls the function for elements that are actually present in the array. If passed our hypothetical array with three elements and a length of 248, it will only call the function three times, not 248 times. If this is how you want to handle sparse arrays, .forEach may be the way to go even if your interpreter supports for...of.

The final option, which works in all versions of JavaScript, is an explicit counting loop. You simply count from 0 up to one less than the length and use the counter as an index. The basic loop looks like this:

var i, s, myStringArray = [ "Hello", "World" ], len = myStringArray.length;
for (i=0; i<len; ++i) {
  s = myStringArray[i];
  // ... do something with s ...
}

One advantage of this approach is that you can choose how to handle sparse arrays. The above code will run the body of the loop the full length times, with s set to undefined for any missing elements, just like for..of; if you instead want to handle only the actually-present elements of a sparse array, like .forEach, you can add a simple in test on the index:

var i, s, myStringArray = [ "Hello", "World" ], len = myStringArray.length;
for (i=0; i<len; ++i) {
  if (i in myStringArray) {
    s = myStringArray[i];
    // ... do something with s ...
  }
}

Depending on your implementation's optimizations, assigning the length value to the local variable (as opposed to including the full myStringArray.length expression in the loop condition) can make a significant difference in performance since it skips a property lookup each time through. You may see the length caching done in the loop initialization clause, like this:

var i, len, myStringArray = [ "Hello", "World" ];
for (len = myStringArray.length, i=0; i<len; ++i) {

The explicit counting loop also means you have access to the index of each value, should you want it. The index is also passed as an extra parameter to the function you pass to forEach, so you can access it that way as well:

myStringArray.forEach( (s,i) => {
   // ... do something with s and i ...
});

for...of doesn't give you the index associated with each object, but as long as the object you're iterating over is actually an instance of Array (and not one of the other iterable types for..of works on), you can use the Array#entries method to change it to an array of [index, item] pairs, and then iterate over that:

for (const [i, s] of myStringArray.entries()) {
  // ... do something with s and i ...
}

The for...in syntax mentioned by others is for looping over an object's properties; since an Array in JavaScript is just an object with numeric property names (and an automatically-updated length property), you can theoretically loop over an Array with it. But the problem is that it doesn't restrict itself to the numeric property values (remember that even methods are actually just properties whose value is a closure), nor is it guaranteed to iterate over those in numeric order. Therefore, the for...in syntax should not be used for looping through Arrays.