In Arabic, the word Tayammum literally means an 'aim' or 'purpose.' In Islamic Law, it refers to: 'Aiming for or seeking soil to wipe one's face and hands with the intention of purification and preparing oneself to pray, and so on." 
Tayammum, is an Islamic ritual to subsidize wudu or ghusl in the absence or impossibility of the usage of water. It is one of the special gifts to this Ummah. The Messenger of Allah (May Allah honour Him and grant Him peace) said (in the meaning): "I have been given five things which no one before me has been given: ... (and he mentioned) The earth has been made for me a place of worship and ceremonially pure, so that when the time of prayer comes any man may pray." Sahih Al Bukhari Vol 1:331
In another narration by Hudhaifah (R): "And its earth has been appointed for us as a means of cleansing when we do not find water." Sahih Al Muslim 1060.In another hadeeth by Abu Huraira (R) reported that the Messenger of Allah (May Allah honour Him and grant Him peace) said: I have been given superiority over the other prophets in six respects: I have been given words which are concise but comprehensive in meaning; I have been helped by terror (in the hearts of enemies): spoils have been made lawful to me: the earth has been made for me clean and a place of worship; I have been sent to all mankind and the line of prophets is closed with me. Sahih Al Muslim 1062. 
1- If one cannot find water or the amount one finds is insufficient for ablution
'Imraan Ibn Husayn (R) said: "We were with the Messenger of Allah (May Allah honour Him and grant Him peace) during a journey. When he led the people in prayer, one man stayed apart. He (May Allah honour Him and Grant Him peace) asked him: "What prevented you from praying?" He replied: 'I need a ritual bath (because of having a wet dream) and there is no water.' The Messenger (May Allah honour Him and grant Him peace) said: 'Use the soil, for it is sufficient (i.e. Tayammum).''' Sahih Al Bukhari and Sahih Al Muslim. For Allah (Glory be to Him) said (in the meaning): "...And you find no water, then perform Tayammum..." Qur’an.Surah Nisa 4:43 and Surah Maidah 5:6.
2- If one is injured or ill
If one is in this condition and thinks that using water will worsen it (he does not have to be absolutely sure, but may base his assumption on past experience or what a knowledgeable person has told him), he may perform Tayammum.
Jaabir (May Allah be pleased with Him) said, "We were on a journey and one of us was injured. Later, he had a wet dream. He asked his companions, 'Can I perform Tayammum?' They said, 'No, not if you have water.' He performed Ghusl, which caused him to die. When news of this came to the Messenger of Allah (May Allah honour Him and grant Him peace) he said: "They killed him! May Allah kill them! Do you not ask if you do not know? The remedy for ignorance is to ask. He could have performed Tayammum and not dropped water on his wound, or wrapped it with something, then wiped over the wrapping, and then washed the rest of his body."" This is related by Abu Dawud, Ibn Majah, ad-Daraqutni
3- If water is cold enough to physically harm the user
Tayammum is allowed when water is available but is too cold and cannot be heated, or there is no public bathroom available. 'Amr Ibn Al-'Aas (R) narrated that he was once participating in a military expedition. He had a wet dream during an extremely cold night, and was afraid that if he performed Ghusl he would die. He prayed the morning (Fajr) prayer with his companions. He then went to the Messenger of Allah (May Allah Honour Him and grant Him peace) to ask him about this. The Messenger (May Allah honour Him and grant Him peace) said: "O 'Amr! Did you pray with your companions while you needed a ritual bath?" 'Amr said to the Prophet Muhammad (May Allah honour Him and grant Him peace) the verse (which means): "…Do not kill yourselves [or one another]. Indeed, Allah is to you ever Merciful." Qur’an.Surah Nisa 4:29. The Prophet Muhammad (May Allah honour Him and Grant Him peace) smiled at this and did not say anything. Sahih Al Bukhari
4- If water is nearby, but one cannot fetch it due to fear
If one fears for his life, family, wealth, [for example, if an enemy (beast or human) is nearby or if one is a prisoner, and so on], he may perform Tayammum. This is also allowed if there is water but one lacks the proper means to get it, or if one fears some accusation against him if he gets it.
5- If one is saving his water for later use
This could be for dough, cooking, or to remove an impurity that is not pardonable. Imaam Ahmad may Allah have mercy upon him said: "Many of the Companions performed Tayammum to save their water for drinking." 'Ali (May Allah be pleased with him) said: "A man who is travelling and becomes unclean because of sexual intercourse, or a wet dream, can perform Tayammum if he fears he will go thirsty. He should perform Tayammum and not Ghusl." Ad-Daaraqutni, Muwatta Malik 94.  
Most scholars agree that the 'clean soil' here refers to at-Turaab (dust of the earth). They say, Tayammum can only be made with at-Turaab. That means, we can't make Tayammum with something that doesn't have dust.