Tayammum is a dry ablution. Basically, it refers to using a clean substance from the earth (soil, sand, dust, etc.) for purification instead of water. Allah says:
(But if you are ill or on a journey or any of you comes from answering the call of nature, or you have been in contact with women (i.e. sexual intercourse) and you find no water, then perform Tayammum with clean earth and rub therewith your faces and hands. Allah does not want to place you in difficulty, but He wants to purify you, and to complete His favor on you that you may be thankful. ) (5:6).
The Prophet (peace and blessing be upon him) said:
"The earth is a place of prayer and means of purification; therefore, when it is time to pray, let one perform prayer wherever he is." (Bukhari).
When can one perform Tayammum?
1. When there is no water available or there is an insufficient quantity for purification. Imran ibn Husain (may Allah be pleased with him) said: 'We were traveling with the Messenger of Allah (peace and blessing be upon him) and he led people in prayer. He saw a man standing aside, and he (peace and blessing be upon him) said: "What prevents you from performing prayers?" He said: "I am in a state of major ritual impurity, and I have no water (to purify myself)." The Prophet (peace and blessing be upon him) said: 'Purify yourself with pure (dry) earth, for it is sufficient." (Bukhari).
The Messenger of Allah (peace and blessing be upon him) said:
"Pure (dry) earth is a (sufficient) means of purification for him who cannot find water for ten years." (Tirmidthi).
2. When the use of water is detrimental to one's health. Jabir (may Allah be pleased with them) said: 'We went on a travel, and one of those who were with us was wounded by a rock on his head. He later on had a wet dream, and he asked his companions: 'Is there any permit for me so that I can perform Tayammum?' His companions said: "We do not see you in a state whereby you can use that permit, since you can use water." He then made Ghusl and died. When they went to the Messenger of Allah (peace and blessing be upon him) and informed him of what had happened, he said:
"They killed him, may Allah kill them. They should have asked when they were ignorant! Indeed the cure of ignorance is asking." (Tirmidthi).
3. When the water is extremely cold and there is no way to heat it. Amr ibn AI-Aas (may Allah be pleased with him) said, 'I had a wet dream during a very cold night, and thought that if I bathed I would kill myself, so I made Tayammum and led my Companions in prayer. When we arrived in Madinah and it was mentioned to him what I had done, the Prophet (peace and blessing be upon him) said:
"O Amr did you lead your Companions in prayer and you were in a state of major ritual impurity?"
He said: "I remembered the words of Allah:
(And do not kill yourselves (nor kill one another). Surely, Allah is Most Merciful to you.) (4:29).
So I made Tayammum and performed the prayer. The Messenger of Allah (peace and blessing be upon him) laughed and said nothing. (Ahmed).
The Prophet (peace and blessing be upon him) approved of what Amr (may Allah be pleased with him) did.
4. When seeking water endangers one's life and property.
5. When water is available but in limited quantity that is needed for drinking, cooking or removing defilement from clothes.