A look to the methods:

• Array#map returns a (new) value for each element.

• Array#filter returns exactly the element if the return value of the callback is truthy

You could take two steps, one for filtering the items and another to get the values from.

const
    students = [{ name: 'Nick', achievements: 158, points: 1473 }, { name: 'Nick', achievements: '175', points: '16375' }, { name: 'Ramon', achievements: '55', points: '2025' }],
    arrayPoints = students
        .filter(student => student.name === 'Nick') 
        .map(student => student.points);

console.log(arrayPoints);

If Array#flatMap is implemented, you could take a single loop and filter and return a value.

The empty array has no items and this array is a neutral value which does not turn up in the result array.

const
    students = [{ name: 'Nick', achievements: 158, points: 1473 }, { name: 'Nick', achievements: '175', points: '16375' }, { name: 'Ramon', achievements: '55', points: '2025' }],
    arrayPoints = students
        .flatMap(student => student.name === 'Nick'
            ? student.points
            : []
        );

console.log(arrayPoints);

For single loop result without undefined. You could do with Array#reduce

students.reduce(function (acc,el) { 
    if(el.name=='Nick'){
        acc.push(el.points)
    }
    return acc
},[]);

You can use reduce for that:

The reduce() method executes a reducer function (that you provide) on each element of the array, resulting in single output value.

So You can use it and check if the name is indeed Nick (el is the currentValue).

if so then push the points to the accumulator (which is arr).

[] represent the initialValue passed to the reduce function.

var arrayPoints = students.reduce((arr, el) => 
    (el.name === 'Nick' && arr.push(el.points), arr), [])  

You can find more info regarding reduce here:

https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/Reduce