You can take Array.map(). This method returns an array with the elements from the callback returned. It expect that all elements return something. If not set, undefined will be returned.
var students = [{
name: 'Nick',
achievements: 158,
points: 14730
}, {
name: 'Jordan',
achievements: '175',
points: '16375'
}, {
name: 'Ramon',
achievements: '55',
points: '2025'
}];
var nameArray = students.map(function (el) { return el.name; });
document.getElementById('out').innerHTML = JSON.stringify(nameArray, null, 4);
<pre id="out"></pre>