The 1095 tax forms notifies the IRS you have had health insurance for 2015. Not having health insurance may result in paying a penalty to the IRS.
If you enrolled through the Federal Marketplace for a premium tax credit 1095 forms were not delayed. The deadline for the 1095 tax form to be mailed was January 31st. If you have not received your form you can call HC.gov at 1-800-318-2596 to inquire. They can also provide you with the tax information you need to complete your taxes.
However if your 1095C or 1095 tax form is coming from an employer or a health insurance carrier the deadline has been extended until March 31st.
If you do not receive your form from either your health insurance carrier or your employer you can file your taxes without it. However if you make a mistake you will need to amend your return to correct it. If you feel you may qualify for one of the available two dozen Affordable Care Act exemptions you can obtain forms and information here. We highly recommended you seek the advice of a tax adviser prior to submitting an exemption form. The rules are very strict with regards to "affordability". Incorrect use of an exemption form can be very costly.