"What are the eyes?"
If you look at the ends of the axle from underneath the vehicle you will see that there are an upper and a lower eye on each side. They have holes about an inch - around 25MM - in diameter into which are pressed the upper and lower ball joints. When the wheel steers back and forth it turns on the ball joints.
It's impossible to get an exact reading of caster from them, as the only precise surface is the inside of the hole, which you can only access by removing all of the outer components. What you can do is get a combination angle protractor and level. (The attached pictures are of a Starret instrument. They're expensive, at around $80 US, but similar ones can be found for half that price.) Make sure that the car is sitting on a level surface. Put the flat surface of the instrument across the upper and lower eyes and rotate the level until the bubble is centered. Then the instrument will show you the caster angle, or the caster angle plus 90 degrees, depending on the instrument.
You want to perform that exercise on both the front and the rear of the eyes, and on both sides. Write down all the readings, and then do it again. after several tries you should be coming up with the same answers every time. Then you will know that you've developed a good and repeatable technique.
Professional shops have instruments of various kinds that attach to the wheel and measure the caster quite accurately, so you can always take the vehicle to one, but for your purposes so far, if you're careful, you should be able to get the angle to within a degree, and that's good enough. You're not building the space shuttle here.
"What are the U joints and pinion shaft?"
The pinion shaft is the input shaft of the differential. The U-joints are the Universal Joints that connect the pinion shaft to the drive shaft, and the drive shaft to the output shaft of the transfer case.
The engine sits a little lower in the back, and everything that's bolted to it is tilted the same way. So the transfer case output shaft angles slightly downward to the rear, and slightly upward towards the front. If both U-joints on the front drive shaft are the same kind, the pinion shaft should be approximately parallel to the transfer case output shaft, which is parallel to the engine crankshaft.
If the rear U-joint of the front shaft is a double one, it's called a constant velocity joint. In that case the pinion shaft of the front differential should be pointed right at the double joint. Any deviation from the proper alignment will cause vibration in the shaft. Two or three degrees is insignificant, and it has a lot to do also with the length of the shaft and a lot of other factors.
If setting the caster angle conflicts with a proper pinion shaft angle, get the caster right first. If you then get a drive shaft vibration, there are ways to deal with it.