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;
    }
});