To just merge the arrays (without removing duplicates)

ES5 version use Array.concat:

var array1 = ["Vijendra", "Singh"];
var array2 = ["Singh", "Shakya"];

array1 = array1.concat(array2);

console.log(array1);

2023 update

The original answer was from years ago. ES6 is fully supported and IE is finally dead. Here's the simplest way to merge primitive and object arrays:

const merge = (a, b, predicate = (a, b) => a === b) => {
    const c = [...a]; // copy to avoid side effects
    // add all items from B to copy C if they're not already present
    b.forEach((bItem) => (c.some((cItem) => predicate(bItem, cItem)) ? null : c.push(bItem)))
    return c;
}

merge(['a', 'b', 'c'], ['c', 'x', 'd']);
// => ['a', 'b', 'c', 'x', 'd']

merge([{id: 1}, {id: 2}], [{id: 2}, {id: 3}], (a, b) => a.id === b.id);
// [{id: 1}, {id: 2}, {id: 3}]

Original answer

ES6 version use destructuring

const array1 = ["Vijendra","Singh"];
const array2 = ["Singh", "Shakya"];
const array3 = [...array1, ...array2];

Since there is no 'built in' way to remove duplicates (ECMA-262 actually has Array.forEach which would be great for this), we have to do it manually. Note that this pollutes the Array prototype, use with caution.

Array.prototype.unique = function() {
    var a = this.concat();
    for(var i=0; i<a.length; ++i) {
        for(var j=i+1; j<a.length; ++j) {
            if(a[i] === a[j])
                a.splice(j--, 1);
        }
    }

    return a;
};

Then, to use it:

var array1 = ["Vijendra","Singh"];
var array2 = ["Singh", "Shakya"];
// Merges both arrays and gets unique items
var array3 = array1.concat(array2).unique(); 

This will also preserve the order of the arrays (i.e, no sorting needed).

Since many people are annoyed about prototype augmentation of Array.prototype and for in loops, here is a less invasive way to use it:

function arrayUnique(array) {
    var a = array.concat();
    for(var i=0; i<a.length; ++i) {
        for(var j=i+1; j<a.length; ++j) {
            if(a[i] === a[j])
                a.splice(j--, 1);
        }
    }

    return a;
}

var array1 = ["Vijendra","Singh"];
var array2 = ["Singh", "Shakya"];
    // Merges both arrays and gets unique items
var array3 = arrayUnique(array1.concat(array2));

For those who are fortunate enough to work with browsers where ES5 is available, you can use Object.defineProperty like this:

Object.defineProperty(Array.prototype, 'unique', {
    enumerable: false,
    configurable: false,
    writable: false,
    value: function() {
        var a = this.concat();
        for(var i=0; i<a.length; ++i) {
            for(var j=i+1; j<a.length; ++j) {
                if(a[i] === a[j])
                    a.splice(j--, 1);
            }
        }

        return a;
    }
});

JavaScript O(N) 131 124 116 92 (86?)

Golfed version:

function m(i,x){h={};n=[];for(a=2;a--;i=x)i.map(function(b){h[b]=h[b]||n.push(b)});return n}

Human readable golfed version:

function m(i,x) {
   h = {}
   n = []
   for (a = 2; a--; i=x)
      i.map(function(b){
        h[b] = h[b] || n.push(b)
      })
   return n
}

I could use concat like so and do it in 86 characters:

function m(i,x){h={};n=[];i.concat(x).map(function(b){h[b]=h[b]||n.push(b)});return n}

But I am not sure if it is still O(N) based on this JsPerf: http://jsperf.com/unique-array-merging-concat-vs-looping as the concat version is marginally faster with smaller arrays but slower with larger arrays (Chrome 31 OSX).

In practice do this (golf is full of bad practices):

function merge(a1, a2) {
   var hash = {};
   var arr = [];
   for (var i = 0; i < a1.length; i++) {
      if (hash[a1[i]] !== true) {
        hash[a1[i]] = true;
        arr[arr.length] = a1[i];
      }
   }
   for (var i = 0; i < a2.length; i++) {
      if (hash[a2[i]] !== true) {
        hash[a2[i]] = true;
        arr[arr.length] = a2[i];
      }
   }
   return arr;
}
console.log(merge([1,2,3,4,5],[1,2,3,4,5,6]));

I'm not great at computing complexity but I believe this is O(N). Would love if someone could clarify.

Edit: Here is a version that takes any number of arrays and merges them.

function merge() {
   var args = arguments;
   var hash = {};
   var arr = [];
   for (var i = 0; i < args.length; i++) {
      for (var j = 0; j < args[i].length; j++) {
        if (hash[args[i][j]] !== true) {
          arr[arr.length] = args[i][j];
          hash[args[i][j]] = true;
        }
      }
    }
   return arr;
}
console.log(merge([1,2,3,4,5],[1,2,3,4,5,6],[1,2,3,4,5,6,7],[1,2,3,4,5,6,7,8]));

You can use concat() and Set together as below,

var a = ["1","2","3"];
var b = ["3","4","5"];
var c = ["4","5","6"];

var d = a.concat(b).concat(c);
var set = new Set(d);

d = Array.from(set);

console.log(d);