Cant hook up HP (Hewlett-Packard) Officejet J4580 All-In-One Printer in order to send and receive fax from standard phone line

Unplug the HP from the phone line. Connect any regular phone, do you have a calling signal? Can you outdial and receive calls? If so... Check if you're plugging the phone cable to the "LINE" not "EXT".

Try Using the HP Print Diagnostic Utility in Windows - HP Customer Care (United States - English).

CN642A - Printhead Unit

This product is compatible with :

Hewlett-Packard OfficeJet 6000
Hewlett-Packard PhotoSmart B8550
Hewlett-Packard PhotoSmart B8553
Hewlett-Packard PhotoSmart C5380
Hewlett-Packard PhotoSmart C6380
Hewlett-Packard PhotoSmart D5460
Hewlett-Packard PhotoSmart Premium Fax
Hewlett-Packard PhotoSmart Premium Fax C309 Series

It is not possible to email the scanned file directly through the printer. However, you may send fax over the Internet

You may subscribe to a low cost phone service that allows you to send and receive faxes with your device by using the Internet. This method is called Fax over Internet Protocol (FoIP). You probably use a FoIP service (provided by your telephone company) if you:
• Dial a special access code along with the fax number, or
• Have an IP converter box that connects to the Internet and provides analog phone ports for the fax connection.

NOTE: You can only send and receive faxes by connecting a phone cord to the port labeled "1-LINE" on the device. This means that your connection to the Internet must be done either through a converter box (which supplies regular analog phone jacks for fax connections) or your telephone company.

Some Internet faxing services do not work properly when the device is sending and receiving faxes at high speed (33600bps). If you experience problems sending and receiving faxes while using an Internet fax service, use a slower fax speed. You can do this by changing the Fax Speed setting from High (the default) to Medium. For information on changing this setting, see Set the fax speed. If you have questions about internet faxing, contact your internet faxing services
support department.

(Ref from the manual of the printer.)

Hope this helps.

is your phone line connected to your machine properly? Is there a dial tone at the jack?

that's right,you would be able to send fax,but wont be able to hear the fax tone,your fax ring shoud be 2 rings higher than answering machine,or else,do 1 thing when you hear the ring,just pick up the receiver u wud be able to hear the fax tone,do not disconnect the call,just keep the receiver aside without disconnecting,and then press start fax...you will be able to receive fax...........

Check this troubleshooting page from HP Support:
HP All-in-One or Fax Product - Troubleshooting Fax Transmission and Reception Issues

Some other device is on the phone line.

Comcast is lying to you. Fax machines are ANALOG devices and will not work reliably on DIGITAL phone lines.

Ok - phone off hook is a message on the printer because you are on the phone. There is a phone that is up somewhere in your home that it can recognize. You cannot send a fax or receive and be on the phone at the same time. This can also be caused by credit card machines, internet service (filter needs to be on fax line), and other fax machines.

To the original poster - your question is fairly vague. This hp link solves the most common issues when you can send but not receive.

http://h10025.www1.hp.com/ewfrf/wc/document?docname=c01503742&lc=en&dlc=en&cc=us&lang=en&rule=21569&product=391182