To properly understand the Spinal Tracts you need to understand the actual anatomy of the spinal cord first. I would recommend Dr. Najeeb's videos on Ascending Tracts but they are pretty long at 3-4 hours (although you can skip through them to get the point).
Take a look at this picture first:
http://www.csuchico.edu/~pmccaffrey/syllabi/CMSD 320/images/U8CrossSec.jpg
What the image doesn't show is that after the "Spinal Nerve" you have another anterior and posterior division. Now before this division the dorsal root carries the sensory fibers and the ventral root carries the motor fibers. All these fibers will meet up in the "Spinal Nerve" of each level of the vertebrae and they will mix and go to their separate ways. So for example, the sensory and motor fibers for the erector spinae muscles of the back would go in the posterior division. I couldnt find a proper image of this (and since it was a source of confusion for me) I drew up an image. Its nothing special, but you can see that the sensory fibers coming from the muscle and the motor going to the muscle meet up before each going their separate ways in the dorsal and ventral roots.
Now I'm sure you also know that Gray Matter makes up the cell bodies while the White Matter is made of the Axons that are going up and down. So now onto the tracts:
Ascending Tracts (i.e. going from down to up) are going to carry Sensory information from the body's various receptors (Pacinian, Meissner's, A-Delta fibers, etc...) to the VPL nucleus of the Thalamus and then to the corresponding sensory cortex area. I remember this from my Neuro classes so I could be mistaken, but sensory tracts have 3 orders of neurons because they go to the thalamus as opposed to motor tracts which only have two. The sensory fibers will also have their cell bodies found inside the Spinal Ganglion, so basically the sensory fibers coming from the body are the dendrites and the pathway going into the spinal cord is the axon.
Descending Tracts are the exact opposite, they go from the motor cortex in the brain and then they will synapse in the
Gray Matter of the spinal cord of the corresponding level (T1/T2 etc...) and then go to the correct muscle to stimulate it. Again, these tracts only consist of 2 orders of neurons because they don't have to pass through the Thalamus.
Now that that is set, we can go onto the 3 major tracts FA mentions.
1) The Dorsal Column/Medial Leminiscus: This tract is responsible for carrying Pressure, Touch, Proprioception and Vibration from the body. As such, think of it as carrying all the information from the bodies receptors, while the Spinothalamic tract carries information from the free nerve endings (page 435, FA 2012). Now after entering the Spinal Cord through the dorsal root (since its a sensory tract) they are going to form 2 tracts in the posterior/dorsal part of the spinal cord. These two tracts are the Fasiculus Gracilis and Cuneatus. Basically the Gracilis, which is located medially, carries the information from the lower half of the body while the Cunateus, which is located lateraly, carries information from the upper half of the body minus the head (which is via the Trigeminal nerve). Think of it as Gracilis carries "Graceful" feet.
Now this pathway is going to go all the way up through the spinal cord until it reaches the medulla where it will synapse on the Gracililis and Cuneatus nucleus. From there the fibers cross the midline and ascend as the Medial Leminiscus. Two things to note here, is that they are now ascending contralaterally to their point of origin since they crossed the midline and they're going to switch around so the gracilis fibers are now lateral and the cuneatues fibers are now medial. Then they're going to rise through the brain stem, go to the VPL nucleus of the Thalamus and then get passed out to the corresponding areas.
2) The Spinothalamic Tract: This tract is responsible for carrying the pain and temperature information from the body (except for the face which is by the Trigeminal). Just like the Dorsal Column its going to have its cell body in the spinal ganglion, but what its going to do is that its going to synapse right away in the gray matter of the spinal cord (as opposed to the Dorsal Column which had its first synapse in the Medulla). From there its going to send out another neuron to the opposite/contralateral side by crossing through a part of the spinal cord called the Anterior Commissure. This same neuron is going to go all the way up till it reaches the VPL of the Thalamus before being passed out to the correct cortex area.
Now the important thing here is that the decussation (or midline crossing) happens immediately at the level of the spinal cord and not up in the medulla. This comes into play when you have damage of the spinal cord. Say you get Brown-Sequard (spinal cord hemisection) at T2. This is going to lead to loss of all contralateral pain and touch below this lesion (because you cut the ascending spinothalamic tract), loss of ipsilateral vibration,pressure, etc.. below (because you cut the ascending dorsal column before it crossed the midline) and loss of all sensation ipsilaterally at that level because you cut the spinothalamic
before it crossed the midline and you cut the dorsal column before it crossed the midline. You're also going to have loss of motor activity/UMN lesion signs below the level of the cut because of damage to the corticospinal tract.
3) Corticospinal tract This one is probably the easiest. Neurons arise from the motor cortex, go to the medulla where they cross the midline and continue all the way down to the right spinal cord level. Then they make their first synapse in the Anterior Horn of the Gray Matter of the Spinal Cord and send out 2nd order neurons to the correct muscle through the ventral root.
So basically the Corticospinal tract is the same as the Dorsal column except its going in the opposite direction. So in both cases, if you have damage above the medulla/above the midline crossing, you're going to have contralateral symptoms. Which is why a left stroke affects the right side of the body. On the other hand, if the damage occurs below the medulala/after the midline crossing, the symptoms are going to be on the same side of the body.
I hope this wasn't overly complicated, but once you get the basic idea everything else becomes a lot easier.