Any compensation from private company if they fire me?

Your employment with a private company is governed by the letter of appointment issued to your by the company and comes under the ambit of Contract Act and terms of the termination of the employments depends on the condition mentioned in the employment contract. If the contract states that the employment can be terminated at any time by issuing a notice period of one month then there is nothing you could do. If there is no provision stating clearly that your employment can be terminated at any time without issuing any notice then ofcourse you can take legal recourse. The first step of which will be to send the company a legal notice.
We would like to know your monthly salary and the designation of your job so that we can clearly determine if you are protected by Industrial Disputes Act or not.
If you are working at a salary less than Rs.10,000 per month and are not working in a supervisory position, you fall under the jurisdiction of Industrial Disputes Act and should file a written complaint to the Assistant Labour Commissioner who will take required legal action against your company.
On the other hand, if your monthly salary is above Rs.10,000 and / or you are at a supervisory position, your employment with a private company is under the jurisdiction of the contract act and will be governed by your employment contract. Please note that civil courts cannot grant reinstatement and the only relief available to you is to recover your legitimate dues from the company inline with your employment contract.