Tayammum should be performed instead of Wudhu or Ghusl in the following seven circumstances:
When it is not possible to procure sufficient water for performing Wudhu or Ghusl.
If a person is unable to procure water on account of old age or weakness, or fear of a thief or a beast, or because he does not possess means to draw water from a well, he should perform tayammum. The same would apply if acquiring water is intolerably difficult. But in this last situation, if a person, inspite of the difficulty, did not perform tayammum, and did Wudhu, his Wudhu will be valid.
If a person fears that if he uses water his life will be endangered, or he will suffer from some ailment or physical defect, or the illness from which he is already suffering will be prolonged, or become acute or some complications may arise in its treatment, he should perform tayammum. However, if he can avoid the harm by using warm water, he should prepare warm water and do Wudhu, or Ghusl when it is necessary.
If a person fears that if he uses water for Ghusl or Wudhu, he will be involved in hardship because of thirst.
If the body or dress of a person is najis and he possesses only as much water as is likely to be exhausted if he does Ghusl or Wudhu, and no water would be available for making his body or dress Pak, he should make is body or dress Pak and pray Namaz with tayammum. But if he does not have anything upon which he would do tayammum, then he should use the water for Ghusl and Wudhu, and pray with najis body or dress.
If a person possesses such water or container which is not permitted to use, like when they are usurped (Ghasbi) he should perform tayammum instead of Ghusl and Wudhu.
When the time left for Namaz is so little that if a person does Ghusl or Wudhu he would be obliged to offer the entire prayers or a part of it after the prescribed time, he should perform tayammum.
Method of performing Tayammum
Striking or keeping both the palms on the object on which tayammum is valid. As an obligatory precaution, this should be done by both the palms together. Wiping or stroking the entire forehead with the palms of both the hands, and, as an obligatory precaution, its two ends commencing from the spot where the hair of one’s head grow down to the eyebrows and above the nose. And it is recommended that the palms pass over the eyebrows as well. To pass the left palm over the whole back of the right hand and thereafter, to pass the right palm over the whole back of the left hand.
source: Imam Ali Foundation