Is it a crime to be a partner in a firm and simultaneously, working in another company.  What if the company in which an employee works finds out that he is running his own business simultaneously.  What are the legal implications?

It depends on the service conduct rule of the Company where the partner is working. 

It shall be appropriate to quit the employment at once (even if notice pay is to be tendered), settle the a/c, obtain work experience/service certificate, relieving letter without adverse comments (rather good comments), form 16, PF accumulation reports, attested copy of PF withdrawal/transfer forms ( to deposit to RPFC by self), NDC/NOC, acknowledgment of company property submitted by employee and having handed over the charge.

This is bad conduct. You have violated the trust and good faith and must not engage in such activities in future.

You have rightly guessed that it is violation of good faith. In your appointment letter company must have described that employee can not engage in any other gainful employment/business. If it happens to be similar kind of business company can charge you for poaching their business and having caused loss.

Employer usually does not tolerate this kind of act and peruse penultimate recourse available to them.  A zealous and vindictive employer can chase the employee for infinite period of time.

You should ease yourself out of the situation as ap.

If the employer is already aware, you must approach competent and experienced lawyer with all records, without any delay. You may refrain from submitting any written document until or unless you are advised by your lawyer.