Whom do I need to contact to get 3rd party insurance of my two-wheeler? What are the advantages of 3rd party insurance? Is it compulsory per law? If I don't have 3rd party insurance, will I get in trouble if I meet an accident?

It happens, the wording is confusing. Don’t get confused, as per any vehicle insurance you will have three parties.

First party means you or the person whose name the insurance is purchased.

Second party is the insurance company.

Third Party is insurance coverage for other.
Any vehicle will have a first and second party.

A person who pays for the insurance and a person who gives insurance.

Third Party insurance in India is compulsory, if accident happens and if someone else got hurt or someonelse car/vehicle, property etc. got damaged, then as per law you are LIABLE TO PAY FOR THE DAMAGE.

In fact, third party risk is made compulsory in India and the Motor Vehicles Act provides that, vehicle should not be used in public place without having insurance policy covering third party risks.

So you might buy a insurance not covering the damage to your life or vehicle while driving.

But it is COMPULSORY to have third party (damage) insurance cover. So in case of any damage to their vehicle, property or the the other person, the insurance will pay.