Use the Array.prototype.push method to append values to the end of an array:

// initialize array
var arr = [
  "Hi",
  "Hello",
  "Bonjour"
];

// append new value to the array
arr.push("Hola");

console.log(arr);

You can use the push() function to append more than one value to an array in a single call:

// initialize array
var arr = ["Hi", "Hello", "Bonjour", "Hola"];

// append multiple values to the array
arr.push("Salut", "Hey");

// display all values
for (var i = 0; i < arr.length; i++) {
  console.log(arr[i]);
}

Note that the push() method returns the updated length of the array.

Update

If you want to add the items of one array to another array, you can use firstArray.concat(secondArray):

var arr = [
  "apple",
  "banana",
  "cherry"
];

// Do not forget to assign the result as, unlike push, concat does not change the existing array
arr = arr.concat([
  "dragonfruit",
  "elderberry",
  "fig"
]);

console.log(arr);

Update

Just an addition to this answer if you want to prepend any value to the start of an array (i.e. first index) then you can use Array.prototype.unshift for this purpose.

var arr = [1, 2, 3];
arr.unshift(0);
console.log(arr);

It also supports appending multiple values at once just like push.

Update

Another way with ES6 syntax is to return a new array with the spread syntax. This leaves the original array unchanged, but returns a new array with new items appended or prepended, compliant with the spirit of functional programming.

const arr1 = [
  "Hi",
  "Hello",
  "Bonjour",
];
const arr2 = [
  "Ciao",
  "Hej",
  "Merhaba",
];

const newArr1 = [
  ...arr1,
  "Salut",
];
const newArr2 = [
  "Salut",
  ...arr2,
];
const newArr3 = [
  ...arr1,
  ...arr2,
];

console.log(newArr1, newArr2, newArr3);

Use unshift. It's like push, except it adds elements to the beginning of the array instead of the end.

• unshift/push - add an element to the beginning/end of an array
• shift/pop - remove and return the first/last element of an array

A simple diagram...

unshift -> [array] <- push
shift   <- [array] -> pop

and chart:

Check out the MDN Array documentation. Virtually every language that has the ability to push/pop elements from an array will also have the ability to unshift/shift (sometimes called push_front/pop_front) elements, you should never have to implement these yourself.

As pointed out in the comments, if you want to avoid mutating your original array, you can use concat, which concatenates two or more arrays together. You can use this to functionally push a single element onto the front or back of an existing array; to do so, you need to turn the new element into a single element array:

const array = [3, 2, 1]

const newFirstElement = 4

const newArray = [newFirstElement].concat(array) // [ 4, 3, 2, 1 ]

console.log(newArray);

concat can also append items. The arguments to concat can be of any type; they are implicitly wrapped in a single-element array, if they are not already an array:

const array = [3, 2, 1]

const newLastElement = 0

// Both of these lines are equivalent:
const newArray1 = array.concat(newLastElement) // [ 3, 2, 1, 0 ]
const newArray2 = array.concat([newLastElement]) // [ 3, 2, 1, 0 ]

console.log(newArray1);
console.log(newArray2);