April 18 is the deadline to either e-file your taxes or postmark your mailed copy, or to file an extension request (in which case you have until October). (This assumes you are a US resident; if you're a non-resident citizen, you may have a different, later deadline.)
IRS Topic 301:
Your return is considered filed timely if the envelope is properly addressed and postmarked by the due date.
Whether or not you submit an extension request, you must submit your payment similarly by April 18th (mail it or e-send it) in order to avoid penalties. Even if you file for an extension, you must send money equal to what you expect to owe approximately in order to avoid underpayment penalties.
When you owe a balance - You may pay any balance you owe via Direct Pay. Alternatively, you may enclose a check or money order with your paper return or, if you filed electronically, with the 2015 Form 1040-V (PDF), Payment Voucher
You will owe interest on any past-due tax and you may be subject to a late-payment penalty if the payment of tax is not made by the original due date of your return.
You will not get a physical bill, unless you fail to pay your owed taxes, in which case you will eventually get the collections department coming after you as with any other unpaid bill. See Topic 202:
If you are not able to pay the tax you owe by your original filing due date, the balance is subject to interest and a monthly late payment penalty. There is also a penalty for failure to file a tax return, so you should file timely even if you cannot pay your balance in full. It is always in your best interest to pay in full as soon as you can to minimize the additional charges.