Redfish started out OK, but he left out the rest
Here's how this works. Your start relay has two push-on connections on any auto trans car, and a stick car with clutch safety
Older cars with a stick only have one push- on connector--the yellow wire
So see if the yellow has power, and the relay clicks.
If the relay does not click, and there is only one yellow push on, the relay is bad
If the relay has TWO push-on connectors, the second wire goes to ground through the neutral safety switch or clutch switch. To check this out, identify which wire for SURE is the start signal by unhooking one or the other, then see which one lights when the key is in crank.
Put this wire (yellow) back onto EITHER of the push-on terminals. Take a clip lead and GROUND the remaining push on terminal on the relay
If the car now cranks using the key, you have a clutch safety switch/ neutral safety switch or related wiring problem
If the relay clicks it MIGHT be bad, might be good.
To find out, jumper the two exposed terminals up on the relay. If the relay clicks "with the key" and no crank, but cranks when you jumper the two exposed relay terminals, the relay is still BAD
If the starter does not crank when you jumper the two relay terminals, either you have a bad wire down to the starter, or the starter is bad.
You'll have to "rig" (clip) your test lamp to the starter battery (large) terminal, to see if you are losing battery voltage as you jumper up at the relay. If so, obviously you have a bad battery cable from the relay to the starter
If the big stud remains hot, move your lamp clip to the SMALL (solenoid) starter terminal. Jumper up at the relay, and if the lamp lights, but no crank, you have a bad starter/ solenoid
Bear in mind that a bad STARTER can also cause the solenoid to "not" pull in
