Bubble sort, quicksort, selection sort, and insertion sort are common sorting algorithms. Bubble sort is one of the simplest but least efficient, comparing adjacent elements and swapping them if out of order. Quicksort chooses a pivot element and partitions the list into sublists based on element values relative to the pivot, then recursively sorts the sublists. Selection sort finds the minimum element and swaps it into sorted position, repeating for remaining elements. Insertion sort inserts elements one by one into sorted position in the list. Sorting algorithms are important for organizing data efficiently and are used in applications like search and output formatting.