Most of the time, dirt comes in the form of grease or oil which sticks itself onto surfaces and will not come off if only water is just used. This is because oil and grease are non-polar, which means that the oil molecules are not charged and therefore are not attracted to polar substances such as water. Because of this, oil tends to stick with its own molecules or other non-polar substances. Oil and grease will stick onto plates and cutlery during cleaning, and no amount of water can completely remove it.
In order to remove oil from clothes or from your dishes with water, the oils must be made soluble by the process of emulsification. Soap is an excellent cleanser because of its ability to act as an emulsifying agent. An emulsifier is capable of dispersing one liquid into another immiscible liquid. This means that while oil (which attracts dirt) doesn't naturally mix with water, soap can suspend oil/dirt in such a way that it can be removed.