Modem connection mysteriously quit

I haven't seen this specifically addressed: I have an HP Pavillion a6742p with an hp external modem attached to the back. Dial-up connection (one computer only) Very slow and very old school but it was working until last night. Now the diagnose and repair msg tells me "A cable is not plugged into the network adapter "Local Area Connection". The phone is fine, so is the fone line. Hooked up my brother's even older computer to internet just fine. Will not even give me a dial tone. External modem blue light is on, no problem with the external wires...I don't have wireless or router, etc., just phone line.
What happened?

"Local Area Connection" is refers to the ethernet (LAN) port on the back of the machine. Since you're presumably using a dial-up modem (through a serial [rs232] port), the offline status of the ethernet connection is expected and I doubt that's your actual problem. In other words, the reason it's saying "a cable is not plugged into..." is because, uh, a cable is not plugged into that socket

The dial tone thing is more worrying. How exactly are you invoking your internet connection via dial-up? When you tested your brother's machine (successfully), did it use your modem during the test or his? Can you swap modems temporarily if you haven't already tried that? There's a possibility that your modem has broken down. All hardware does that, eventually.

There is an error message 680 "no dial tone". I've always just signed on and automatically the connect to internet box comes on. My brother's machine (this one) is older and modem is built in. I just disconnected the phone line from phone to this computer. There is a splitter so both fone and computer can share. I checked the splitter and it's ok too. The HP didn't come with a modem so I had to order from HP separately. I just bought it a couple months ago! ($40) There is a little blue light that goes on when it's plugged in and it's still on. Also, my virus program says all is ok. Could something have "re-configured" itself?

That error message (680 - lack of a dial tone, supposedly) comes to the OS from the combination of the modem hardware and the modem driver. If you know the exact make and model of the modem, you should try downloading the latest driver from the manufacturers website and installing that.

The other thing you could try as a test is booting to [safe mode + net] and attempting the internet connection from there. If that works, it tells us there's something borked in your normal boot configuration that's bypassed when you boot to safe mode.

I've never booted in safe mode before. ??? If needed, can I download a driver on this older computer and save to memory stick for later transfer to non-connected computer?
Thank you for your time. I really am lost! I thought Vista would be more "user friendly", as in, telling me exactly what is wrong. Alas!

Booting to safe mode is accomplished by pressing F8 during the earliest stages of the boot process - as soon as you get past the BIOS and hardware info. In this case you need to choose "safe mode with networking" because you want a bit of networking support to load up too so you can test whether the modem connection works.

Yes, you can download the driver on another computer and copy it across via USB.

About the user-friendliness, it depends where the fault is. The OS itself is just a program. A big one, certainly, but it still requires assistance from the drivers (also software) which directly interface with the hardware. Your driver+hardware (modem) combination is reporting "no dial tone", so that's what reaches the user as an error message.