There are two general ways to do contract work for a company:
1. You can be a direct contractor under your SSN. You file a W-4 with the client company which identifies you and the company then reports all payments at the end of the year on a 1099-MISC. You report your income on Schedule C as a sole proprietor.
2. Your S-corp can contract with the client company under your S-corp EIN. In this case, the company is contracting and paying the S-corp, not you, and is not generally required to file a 1099-MISC since it is just a payment between two corporations. Your S-corp then sends you to the client company as an employee of the S-corp.
Since you are working for your S-corp, the S-corp must pay you a wage, withhold payroll taxes, and file a W-2 for your wages. The S-corp may also pay a dividend to you which requires filing a K-1.
Note that the S-corp is a separate legal entity from you. That is the whole point. You work for the S-corp and are hired out as an employee of the S-corp. You may also be the only employee of the S-corp and the boss of the S-corp, but you are still an employee.
If you are just doing hourly contract work for a company, you may make your life a lot simpler by just skipping the whole S-corp thing and just work directly as a sole proprietorship as in case 1 above. You only need to form an S-corp if your client insists on dealing with a corporation rather than an individual contractor. You should check with your client and see if direct 1099 contracting is okay and save yourself a lot of grief, time and expense by avoiding an S-corp.