Dom is the active partner in D/s. It is often assumed to be a shortened version of the word 'dominant', used as a noun but could equally come from the Latin Dominus (man of the house) or Domina (lady of the house). It is usually written with an upper-case D as an honorific.
People are clearly entitled to call themselves whatever they want and the term Dom or Dominant is sometimes mixed up with both Top and Master. Indeed, a Top might be dominant and a Master almost certainly is.
A Dom is a partner who takes the role of giver or controller in such acts as bondage, discipline, humiliation, or servitude. Such acts are performed on a sub. Many Doms incorporate all aspects of being a Top but this is not universal.
While a Dom will take care of his sub, he will be giving orders or otherwise employ physical or psychological techniques of control. He might instruct the submissive to perform the act on him.
The relationship between a Dom and his sub may be very transitory or can be permanent. Some Doms are married to their sub. Long term relationships are known as D/s (Dominant/submissive) relationships.
Some partners switch roles from one encounter to the other, or even during a single encounter, depending on mood and preference. See Switch.