Modifying arrays from an input then returning an output

The biggest problem is that you never access the elements of input, just the array itself. Your if (input % 2 == 1) line checks if the array mod 2 is equal to 1. That doesn't make sense, as you can't really do math on an array.

You need to do this per-element, so the minimal change would be to use input[i] within the loop.

There are a number of other changes you can make, and some idiomatic patterns that can make the code nicer. If you're on a recent browser and have forEach or map, you can replace the loop with a construct like:

var output = input.map(function (it) {
    if (it % 2 == 1) return it * 2;
    if (it % 2 == 0) return it / 2;
});

That can still be cleaned up, as x % 2 can only ever return 0 or 1, so you can replace the second condition with an else or just assume it returned 0:

var output = input.map(function (it) {
    if (it % 2 == 1) return it * 2;
    return it / 2;
});

Because of how JS handles true and false, specifically converting numbers, you can omit the == 1 (1 is truthy), and swap the condition for a ternary:

var output = input.map(function (it) {
    return (it % 2) ? it * 2 : it / 2;
});

Since you're wrapping this into a function, if you're using map then output isn't strictly necessary, so you can do:

module.exports.modify = function (input) {
    return input.map(function (it) {
        return (it % 2) ? it * 2 : it / 2;
    });
};

If you have ES6 support (likely through the brilliant 6to5 project), you can replace the function declarations with arrow functions:

module.exports.modify = (input) => {
    return input.map((it) => {
        return (it % 2) ? it * 2 : it / 2;
    });
};

And if you want to get really academic with it, you can remove the return statements (thanks to arrow functions):

module.exports.modify = input => input.map(it => it % 2 ? it * 2 : it / 2);

This is wrong:

for (i=0; i=input.length; i++) {
      if (input % 2 == 1) {

input is the entire array. You can't modulo an array by 2.

You need to check individual elements with the [] operator:

if (input[i] % 2 == 1) ...

The same applies inside the bodies of the if condition:

        input*2;
        output.push[input];

This needs to be input[i] * 2, and you also need to store that value. Right now, you have two unrelated statements: input * 2, which (if it were correct) would compute the value and discard it. Then (and again, your syntax is in correct, you need () not [] here) output.push[input] would append the original value to output.

[] is for accessing an element in an array, and () is for invoking a function. push is a function. The line could be correctly written as:

output.push(input[i] * 2)

A few errors here;

Your for loop condition makes sure that i is equal to the length of the array, not less than it. Unless your input array is empty the loop will never run.

for(var i = 0; i < input.length; i++)

You're checking if input (the array) is divisible by 2, not if the element inside of input is.

if(input[i] % 2 === 1)

You're also performing a math operation against the array object, not the element in the array, and you're not persisting it anywhere.

var multiplied = input[i] * 2;

You're calling the push method of the output array as an array, not as a function call.

output.push(multiplied);