The way I understand one of the way LDAP works is that, and correct me if I am wrong, a user can sign into it from his computer and access a bunch of web applications provided he has the permission to do so and the credentials match.
I am creating a web application and I want to configure it with LDAP. So, I should, based on my client, make use of the LDAP API and ensure that he has the required permission to do so. Then
You are mixing up several things.
Applications (any type of applications) use different authentication techniques. In the Java web world, there is something called JAAS.
The Java Authentication and Authorization Service (JAAS) is a set of application program interfaces (APIs) that can determine the identity of a user or computer attempting to run Java code, and ensure that the entity has the privilege or permission to execute the functions requested... (Wikipedia)
In .NET, you have a framework called WCF which has a dedicated section on security.
With that in mind, let's get back to your question. If you have a Java web-app and you use LDAP as a user directory, then you will most likely be achieving the integration between your app and LDAP via JAAS.
Will LDAP send your app the password? NOOOOOOOO never, that would a terrible idea. Hopefully, by the way, your LDAP does not store the password in the clear. Typically passwords would be one-way encrypted (hashed) i.e. it is impossible to retrieve the value of the password from the encrypted value.
So, what can LDAP (via JAAS) tell you?
One last thing. Usually authentication (and authorization) are not directly handled by the application but rather by the application's container. If you write a Java web-app, odds are that you will run it inside Apache Tomcat. Tomcat comes with JAAS and has native integration with Tomcat's internal user store as well as others (e.g. LDAP again or a database).
So your first question should be: what language am I writing the app in? How will I run it? Do I get anything for free?
Here is an old Oracle tutorial on securing Java web-apps.