All soaps contain basically the same chemical, sodium hydroxide, NaOH (a base). You might have learned in chemistry classes that an acid + a base combine to make a salt + water. The resulting salt is what we call soap. One part of the soap molecule is hydrophilic (water-binding) and the other is hydrophobic (water-repellent). The hydrophilic part allows the hydrophobic fatty acids to come into contact with other hydrophobic substances, such as the dirt on the surface that is being cleaned. When the grime adheres to the soap's fatty acids, it becomes encapsulated in droplets of water. Dirt, oil and bacteria are easily scrubbed off and washed away in this suspended state.

So there's really only one kind of basic soap. The different kinds you see in the store have different additives, such as perfumes or anti-bacterial agents, but the soap part acts in the same way. (Notice however that detergents are something different again!)