I was exempted from the federal tax last year from my employer. When I am filling my tax it shows I'm owing tax. How can I take care of my exemption?

Why do you think you are exempt?

My employer told me that I was exempted.

Did you get a W-2, or were you an independent contractor?

I got a W-2, but my federal tax box states nothing.

Your employer does not decide or determine whether or not you are exempt from paying taxes.  Since you are not exempt from income taxes, you should give your employer an updated W-4 that reflects this information.  Your employer should be withholding income taxes and those taxes should be reported in Box 2 of your W-2.  If your employer has not been withholding and you earned over 12K and do not have a qualifying dependent, you will probably owe income taxes.  

You can prepare a new form in TurboTax to give to your employer by typing in W-4 in the search box and using the 'Jump to' link.  Or you can request a blank form from your employer and update it there.  

I filled W-4 last year and apply for exempt as I was meeting all the conditions. Now on my w-2 I don't have any thing in box-2.

If you owe taxes now then you did not meet all of the conditions.  How much is your income and do you have dependents?  Here is the new form W-4 you can fill out for 2019.  There is nothing you can do about 2018 now except pay your taxes unfortunately.  <a rel="nofollow" target="_blank" href="https://www.irs.gov/pub/irs-pdf/fw4.pdf">https://www.irs.gov/pub/irs-pdf/fw4.pdf</a>    And here is a chart that shows whether or not you have to file/pay tax. -  <a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/1901490-do-i-need-to-file-a-federal-return-this-year">https://ttlc.intuit.com/questions/1901490-do-i-need-to-file-a-federal-return-this-year</a>