In terms of importance and widespread usage, the following algorithms are considered fundamental and essential to know:

1. Binary Search: A divide-and-conquer algorithm used to efficiently find the position of a target value in a sorted array.

2. Sorting Algorithms:

   • Bubble Sort: A simple comparison-based algorithm that repeatedly swaps adjacent elements if they are in the wrong order.
   • Merge Sort: A divide-and-conquer algorithm that recursively divides the array into two halves, sorts them, and then merges the sorted halves.
   • Quick Sort: Another divide-and-conquer algorithm that selects a pivot element and partitions the array into two sub-arrays based on the pivot.

3. Depth-First Search (DFS): A graph traversal algorithm that explores as far as possible along each branch before backtracking.

4. Breadth-First Search (BFS): A graph traversal algorithm that explores all the vertices of a graph in breadth-first order, i.e., it visits all the neighbors of a vertex before moving to the next level.

5. Dijkstra's Algorithm: A graph algorithm that finds the shortest path between nodes in a weighted graph.

6. Dynamic Programming: A technique to solve complex problems by breaking them down into simpler overlapping subproblems and storing their solutions to avoid redundant computations.

7. Knapsack Problem: An optimization problem that seeks to maximize the total value of items in a knapsack (with a certain weight capacity) while staying within the weight limit.

8. Prim's Algorithm: A minimum spanning tree algorithm that finds the subset of edges that connect all vertices of a weighted undirected graph while minimizing the total edge weight.

9. K-means Clustering: A popular unsupervised machine learning algorithm that partitions a dataset into k clusters based on their similarity.

10. A * Search Algorithm: A heuristic search algorithm often used in pathfinding and graph traversal problems, finding the shortest path between two points.

These algorithms cover a range of fundamental concepts and problem-solving techniques and can provide a strong foundation for programmers in various domains. Remember that there are many more algorithms to explore based on specific needs and interests.