There are several processes for making soap. The two predominant types are cold process and hot process.
Nearly all the soap we make is done by cold process, which simply means lye is added to water, milk, aloe, or a combination, and then this is added to oils and/or butters. Once mixed together the lye water and oil goes through a process called saponification, which means soap has been created!
Hot process is similar in that lye water and oils are combined, but it is done with additional heat. Either way, you end up with soap. This is the way we make shaving soap because of the additional heat required to melt all the ingredients.
Saponification occurs more quickly with hot process, but both require several weeks of curing so that excess water can evaporate. Six to eight weeks is a normal cure time, unless you're talking about true Castile soap, in which case you're probably looking at a full year!
The longer a bar of soap cures, the harder it becomes. That means it will last longer. It will also be more gentle. So, even though it is perfectly safe to use a new bar of soap after 48-72 hours, it will last much longer and you will be able to enjoy it way more if it is fully cured. It also helps if a bar is left in an area where it can dry out completely between uses...such as a draining soap dish or hanging mesh bag.