Currently, I am running my payroll. I got stuck in the tax payments. Could anyone briefly explain me the procedures and policies of the payroll system?

What county? If the US, what state?

Again if US, 1099 employees or W2,

Are there benefits involved, and if so, how are they funded?

And really no, there is no brief way to actually explain it. Some computer systems do a fairly good job at tracking and doing the withholdings.

But you need to hire someone, to come in and show you how it works and to be sure you are set up properly.

https://www.fasb.org/home