Why do we need better algorithms?

This is the most fundamental question every student must ask himself before taking a course on Algorithms. Rather than talking too much about algorithms let me straight away give examples. All the shown examples are from the speech 'Beyond Computation - Michael Sipser'
Problem 1: Finding the factorial of a large number.

We still don't know whether there is any other possible way to find the solution or not.

Problem 2: Finding the largest clique in a given graph with 100 nodes

But trying to find a large clique in the below figure is very difficult for the computer.

Solution: Trying to find the analogy of magnet in mathematics

If you can find the magnet, you are one of the most Intelligent on the planet.

Other Problems:

So you can see that the biggest challenge is 

This is the ultimate reason why we need better algorithms.

To understand more, let me show you the difference between P and NP problems

If P = NP is proved, then we do not need to search for a search algorithm else we need to search. Whether P = NP or P != NP is a mystery.

NP-Completeness

This is the first step taken by humans towards solving an NP problem. NP-Completeness defines that a set of problems are linked to each other and can be easily transformed into each other which means if there is a solution to one problem, then the solution for the other can be found but the reverse is not true. Eg: Factoring can be converted into CLIQUE but not CLIQUE into Factoring

NP-Hard:

There are NP problems for which even checking is not possible. These are called NP-Hard. Eg of such a problem is Chess.

Will it ever be solved?