How will I be penalized if I don't buy health care coverage under ObamaCare?

The IRS will receive data from the Dept of Health and Human Services (the department charged with administering the ACA) to indicate that a person (or family) has not obtained proper coverage.

When you file your taxes, the IRS will then add the appropriate penalty to your taxes, requiring you to either pay more, or reduce the refund owed you.

Despite common lore, the IRS can most certainly enforce this tax penalty. The specific actions that the IRS is allowed to perform to collect this debt are more limited than for a normal tax debt (i.e. they can't seize property, but they CAN garnish wages), but the IRS nonetheless can collect on it.