1. Introduction to Quicksort

Quicksort is a divide-and-conquer algorithm. This means it breaks down a problem into smaller sub-problems, solves each sub-problem, and combines the solutions to solve the original problem. In the world of sorting algorithms, Quicksort is known for its impressive average-case performance.

2. How Does Quicksort Work?

To truly appreciate the simplicity and elegance of Quicksort, let’s understand its primary steps:

2.1. Partitioning

The primary objective of this step is to choose a 'pivot' element from the array and partition the other elements into two sub-arrays, according to whether they are less than or greater than the pivot.

Steps involved in partitioning:

  1. Choosing a Pivot: The choice of the pivot element can be random, fixed (like the first element, the last element, or the middle one), or based on certain heuristics.
  2. Reordering the Array: Once the pivot is selected, other elements are reordered. Those less than the pivot come before it and those greater than the pivot come after it. This operation effectively places the pivot in its correct position in the sorted array.

2.2. Recursive Sort

Post-partitioning, the pivot is in its rightful position. Now, the elements to the left and right of the pivot haven't been sorted. They form the sub-arrays which are then sorted recursively using the same approach.

3. Python Implementation

Here's a simple implementation of Quicksort in Python:

def quicksort(arr):
    if len(arr) <= 1:
        return arr
    pivot = arr[len(arr) // 2]  # Using middle element as pivot
    left = [x for x in arr if x < pivot]
    middle = [x for x in arr if x == pivot]
    right = [x for x in arr if x > pivot]
    return quicksort(left) + middle + quicksort(right)

print(quicksort([3, 6, 8, 10, 1, 2, 1]))

When the above code is executed, it will output: [1, 1, 2, 3, 6, 8, 10].

Note: This is a high-level and illustrative Python implementation. In practice, you'd want a version that sorts in-place without using additional memory for creating left, middle, and right arrays.

4. Advantages and Limitations

4.1. Advantages:

  1. Fast on Average: On average, Quicksort runs in O(n log⁡ n) time.
  2. In-Place Sorting: With a good implementation, Quicksort doesn't require additional storage.
  3. Universally Useful: Works well for various types of input data.

4.2. Limitations:

  1. Worst-case Scenario: In its basic version, the worst-case time complexity can degrade to ᴼ⁽ⁿ²⁾ if the pivot choice is unlucky. However, this can often be avoided with smarter pivot-picking strategies. 
  2. Not Stable: By default, Quicksort isn't stable. This means equal elements might get their original order in the array disturbed.

5. Conclusion

In conclusion, Quicksort is an elegant and highly efficient sorting algorithm that deserves a place in every programmer’s toolkit. Its divide-and-conquer approach not only ensures fast sorting on average but also provides insights into problem-solving techniques that are broadly applicable in computer science.