Physicists can explain a lot of things that happen in the universe, like existence of energy, the bending of space-time etc by using a handful of particles and forces. The standard model predicted that everything in the universe should be massless, but we know that’s not true. So Physicists such as Peter Higgs and Tom Kibble came up with the Higgs mechanism responsible for the generation of massive particles. And in 2012 the Large Hadron Collider found the Higgs Boson which proved the existence of the Higgs field and validated the Higgs Mechanism.
Although we can explain a lot of things with the Standard Model but it is still incomplete. SuSy is an extension of the standard model which aims at explaining some of the phenomenon which is left unexplained by the standard model, just like the Higgs mechanism.
Supersymmetry is not a theory but a principle which can be applied to any theory. A theory can be proved wrong by experiment but Supersymmetry itself cannot be proved wrong.
If you take a spherical ball and rotate it in any possible way, the end result is still the same. The sphere is said to be symmetric under rotation. Similarly, if an equation remains the same if we transpose force terms with matter terms or vice versa, the equation is said to be supersymmetric.
A Supersymmetric theory is the one which treats the equations for FORCES and the equations for MATTER identically.
It turns out if the universe has this property a lot of problems which we face in the standard model are elegantly solved. Three of the most important problems solved by SuSy are-
1. The mass of the Higgs Boson
SuSy predicts that for every fundamental particle that exists there exists a supersymmetric cousin called a Sparticle which has a higher mass than the particle itself.
An electron would have a supersymmetric cousin called a selectron which would have the same charge and spin but higher mass than an electron.
The interactions between the Higgs boson and other standard model particles predict that the Higgs Boson should be a very heavy particle contrary to what we found at the Large Hadron Collider.
If SuSy is true then the extra particles predicted by SuSy would cancel out the mass contribution to the Higgs Boson and therefore explain its low mass.
2. Unification of the forces
Physicists have long wanted to unify the forces of nature into one fundamental force which describes all the phenomena that takes place in the universe. Ever since humans existed we saw the moon moving across the sky and objects falling down to earth when thrown in the air.
Isaac Newton unified these two phenomena into the one force of Gravity. Similarly James Clark Maxwell unified Electricity and Magnetism into Electromagnetism by showing that the two forces are just different consequences of the same basic principles.
The Strong and Weak nuclear interactions along with Electromagnetism have different strengths at all energies.
At very high energies, similar to the conditions in the early universe, the strengths of these three forces take similar shape but do not converge at a single point.
However if SuSy is true then the three forces converge at a single point rather elegantly. A theory that unites the forces mathematically is called a Grand Unified Theory, a dream of physicists including Einstein. Therefore even though there has been no experimental evidence for SuSy, there are more than ten thousand scientific papers written about it.
3. Dark Matter candidate
SuSy not only has application in the micro world of particle physics but it provides a solution to one of the biggest problems in cosmology. Astronomers have known for a long time that the galaxies are spinning much faster than they should spin given the amount of mass that they contain. So to save astronomy and solve this problem, Dark matter was hypothesized.
Dark matter fills the entire galaxy and provides enough gravitational tug to everything which explains the high rotational velocity of the galaxies. It prevents the stars from escaping the galaxy tangentially.
The problem with Dark matter is that it has never been experimentally observed. However, a class of particles predicted by SuSy called LSPs ( Lightest Supersymmetric Particles) have the perfect properties to be the candidate for the Dark Matter particle.
Neutralino, the supersymmetric cousin of the neutrino is the most likely candidate.
Therefore, if SuSy is true it will solve problems both of the quantum and the cosmic scales. The Large Hadron Collider at Cern is in its Second run and one of the things its is looking for is the existence of supersymmetric particles.