These are also known as the Four Bases of Success.
The Buddha gives a very clear analogy of what they mean. Suppose the desire arises in us to go to the park. This is the first iddhi. Then we investigate: should I go or not? we investigate the pros and cons of going to the park. This is the fourth iddhi. Then we make up your mind: yes, I'm going to the park. This is the third iddhi. Finally, we arouse the energy to get to the park. Once we arrive at the park the iddhis subside as they are no longer needed.
In our meditation practice, we use the iddhis to realized the Four Right Efforts:
1. The effort to not allow unwholesome things to arise.
2. The effort to let go of unwholesome things that have arisen.
3. The effort to cause the arising of wholesome things.
4. The effort to maintain and increase the wholesome things that have arisen.
So the first iddhi, desire is the desire to realize the results of these four efforts. We use the fourth iddhi, investigate to consider, "Do I want to realize the Four Right Efforts? What are the pros and cons? The third iddhi, mind, is making up our mind, "Yes, I'm going to make the effort to practice the Four Right Efforts". Then we employ the second iddhi, energy, to arouse the energy needed to practice the Four Right Efforts.
The way leading to the iddhis is the Noble Eightfold Path.
We develop each of the iddhis in the following manner:
1. So they are neither too slack or too tense i.e. not accompanied by lassitude or restlessness.
2. So not constricted internally i.e. no sloth and torpor.
3. So not disturbed externally i.e. not disturbed by the five cords of sensuality i.e. by sights, sounds, tastes, smells or touch/physical sensations.
4. Penetrate before and after making effort with wisdom i.e. understanding cause and effect and the Four Noble Truths.
5. Review the 32 parts of the body from the toes to the head and from the head to the toes.
6. Develop the iddhis during the day and the night.
7. Develop a mind imbued with luminosity i.e. we sit outside and perceive the daylight.
Then we close our eyes and try and see the same amount of light with eyes closed. Practice until this happens.
The reason we use the iddhis to practice the Four Right Efforts is to Awaken to the end of suffering. When the iddhis are developed and the taints destroyed, we Awaken via wisdom. The taints are attachment to sensuality and existence and ignorance of the Four Noble Truths and the law of cause and effect (Dependent Arising). A side effect of the iddhis is that psychic powers can arise.
1. We can clone ourselves into many copies of yourself.
2. We can appear and disappear.
3. We can pass through a wall, a rampart or a mountain.
4. We can dive in and out of the earth.
5. We can walk on water.
6. We can travel through space like a bird.
7. We can stroke the sun and the moon with our hand.
8. We have mastery with the body as far as the Brahma world. (There are six heavenly realms that have some degree of form, albeit, less substantial than our human realm form before the Brahma world realms which are the start of the formless heavenly realms. There are 20 formless realms.)
9. We can hear divine (i.e. can hear the devas speaking) and human sounds and can hear sounds from far away.
10. We can determine the mind states of others.
11. We can recollect our past lives.
12. We can see where beings, who pass away, are reborn, according to the kamma.
Chapter 51 of the Samyutta Nikaya is devoted to the Buddha's teaching on the iddhis. You can read these suttas on suttacentral.net