What is social security tax and medicare tax and why don't I get those back?

They are payroll taxes if you are an employee and are self employment taxes if you are self employed.  They fund the social security and Medicare plans. 

Social Security and Medicare tax fund these entitlements, and there currently is no way to get these taxes refunded. Essentially, you are paying these taxes so you will have Social Security and Medicare available when you reach 62 and 65, respectively. You are paying to fund a portion of your own future retirement (Social Security) and healthcare (Medicare). 

@annaleah2

That's the theory that is used to lull you into complacency.

In actuality, your money is just taken by the government and reallocated to others that they have declared "entitled to it".