I have an '84 T-Bird. Six weeks ago I had the battery and its cables and some other hardware replaced.
A few days ago, I got a click when I tried to start the car. I tried this several times and finally lifted the hood; the cables fastened to the battery terminals seemed unusually warm.
Any Dopers have an idea what the problem is? 
For starters (howzat for a pun?) the problem appears to be in your charging or starting system, not ignition.
Why did you have the repair work done? What was the first problem? If you turn on the headlights, do they shine brightly? What happens when you turn the key-do they go out or continue to shine?
Warm cables are indicative of current flow-that’s all.
I’d start by load testing the battery, evaluating the charging system, and then seeing if starter draw is within acceptable specs.
Its diagnosis, not prognosis 
I’d say that you have some problem with your starter motor and not with the battery (since the cables warmed up).
You can find the starter motor in the engine bay and hit it with something heavy. This works sometimes.
I had the dome light on; there wasn’t anyone else around to look at the headlights. But when I turned the key and got nothing but the click, the dome light did not dim.
Incidentally, I had the starter replaced a few years ago, along with the flywheel (this cost me about $400!) when a broken cog on the flywheel was keeping the starter from operating. I hate to think there is something wrong with the starter (or the solenoid) again… 
Gotcha. Let’s not go for doom and despair just yet, as there is one issue unique to Fords.
Lift the hood and follow the positive lead away from the battery. It will terminate on a black block about 3 inches square mounted to the fender well. Two large diameter bolted connections will be facing you, once you’ve found it, and a smaller diameter wire will sit off to one side.
This part is called a starter relay, and it takes a small voltage input from the ignition switch and closes high amp contacts to make the starter run.
Have someone turn the key and if you hear it clicking but there is no starter action, you can replace it or go on to one more radical step.
Disclaimer: This can be very dangerous unless you’re of steady hand. I’ve done it to start my truck, but folks will argue about my sanity. 
Use a pair of pliers (preferably old) to short the large threaded contacts on the starter relay while the ignition switch is in the ‘run’ position. The starter should immediately crank and start the vehicle. If you get an arc and nothing, stop-as it is time for either starter replacement or invitation of more learned minds.
This sounds rather dangerous: should he be wearing rubber gloves at the time?
Nope.
There’s a lot of amperage there, but only 12 volts and DC to boot. It makes an impressive spark, but can’t hurt you.
As well, the best path for the power is through the pliers, to the starter and back through the car’s body ground to the battery. The path through the pliers, through you, your shoes, the floor, the car’s tires, the car’s running gear, the car’s body and finally back to the battery is MUCH higher resistance and will carry very little, if any of the current flow.
Leather would be better. The high current flow can cause things to get warm. Eye protection is MANDITORY. However it is spelled.
I’d just replace the thing to test this, it is only maybe fifteen bucks.
'Tis true, except that those of us who are possessors of only one vehicle need a means of getting to the auto parts store. Plus, when you’ve had the same truck for 15 years, these little workarounds to common problems are second nature.
Well, if push comes to shove, I can take a bus on the arterial street a half-block away, to a Pep Boys where I could buy a starter relay.
I’ve also noticed that the dome light has come on after dark, two days in a row–and, so help me, I haven’t touched the light switch…
Thanx. 
A neighbor who has connections with independent mechanics contacted one while I was away. He did some work on the starter relay, I assume; he’s charging me $50 all told and I doubt he’s selling starters out of the trunk of his car. 