Tayammum is dry ritual purification which can be done in place of wudu or even ghusl, if there is a valid excuse for that.
Tayammum is permissible as the replacement for wudu and ghusl in cases, when it is impossible or unduly difficult to perform them, such as:
Being roughly 1.86 km (1.16 mile) away from water
Extreme cold, such that one is likely to get ill as a result of using water, and has no means of heating it up
Sickness, whereby one reasonably expects that using water will worsen his health or delay the recovery
Wounds on most of the body
Fear of thirst, such that the only water available is for drinking
Tayammum can be performed with any pure substance of the earth, such as dirt, sand, limestone, stones, or dust.
Tayammum consists of following actions:
Make an intention to remove the state of ritual impurity
Strike the earth once and wipe the face completely with dust (or whatever substance is being used)
Strike the earth a second time and wipe the hands up to and including the elbows completely
After performing tayammum, one becomes pure, even if he was in a state of major ritual impurity prior to that. One remains pure and can perform prayers until his wudu or ghusl are nullified, or the initial excuse making the tayammum permissible is no longer there.