What actually happens when I join a windows machine to an Active Directory domain

Active Directory is a database based system that provides authentication, directory, policy, and other services in a Windows environment

LDAP (Lightweight Directory Access Protocol) is an application protocol for querying and modifying items in directory service providers like Active Directory, which supports a form of LDAP.

Short answer: AD is a directory services database, and LDAP is one of the protocols you can use to talk to it.

This article helped me much to understand how to work with the Active Directory.
Howto: (Almost) Everything In Active Directory via C#

From this point forward, if you require further assitance, please let me know with proper questions in comment, and I shall answer them for you to the best of my knowledge.

EDIT #1

You had better go with this example's filter instead. I have written some sample code to briefly show how to work with the System.DirectoryServices and System.DirectoryServices.ActiveDirectory namespaces. The System.DirectoryServices.ActiveDirectory namespace is used to retrieve information about the domains within your Forest.

private IEnumerable<DirectoryEntry> GetDomains() {
    ICollection<string> domains = new List<string>();

    // Querying the current Forest for the domains within.
    foreach(Domain d in Forest.GetCurrentForest().Domains)
        domains.Add(d.Name);

    return domains;
}

private string GetDomainFullName(string friendlyName) {
    DirectoryContext context = new DirectoryContext(DirectoryContextType.Domain, friendlyName);
    Domain domain = Domain.GetDomain(context);
    return domain.Name;
}

private IEnumerable<string> GetUserDomain(string userName) {
    foreach(string d in GetDomains()) 
        // From the domains obtained from the Forest, we search the domain subtree for the given userName.
        using (DirectoryEntry domain = new DirectoryEntry(GetDomainFullName(d))) {
            using (DirectorySearcher searcher = new DirectorySearcher()){
                searcher.SearchRoot = domain;
                searcher.SearchScope = SearchScope.Subtree;
                searcher.PropertiesToLoad.Add("sAMAccountName");
                // The Filter is very important, so is its query string. The 'objectClass' parameter is mandatory.
                // Once we specified the 'objectClass', we want to look for the user whose login
                // login is userName.
                searcher.Filter = string.Format("(&(objectClass=user)(sAMAccountName={0}))", userName);

                try {
                    SearchResultCollection  results = searcher.FindAll();

                    // If the user cannot be found, then let's check next domain.
                    if (results == null || results.Count = 0)
                        continue;

                     // Here, we yield return for we want all of the domain which this userName is authenticated.
                     yield return domain.Path;
                } finally {
                    searcher.Dispose();
                    domain.Dispose();
                }
            }
}

Here, I didn't test this code and might have some minor issue to fix. This sample is provided as-is for the sake of helping you. I hope this will help.

EDIT #2

I found out another way out:

1. You have first to look whether you can find the user account within your domain;
2. If found, then get the domain NetBIOS Name; and
3. concatenate it to a backslash (****) and the found login.

The example below uses a NUnit TestCase which you can test for yourself and see if it does what you are required to.

[TestCase("LDAP://fully.qualified.domain.name", "TestUser1")] 
public void GetNetBiosName(string ldapUrl, string login)
    string netBiosName = null;
    string foundLogin = null;

    using (DirectoryEntry root = new DirectoryEntry(ldapUrl))
        Using (DirectorySearcher searcher = new DirectorySearcher(root) {
            searcher.SearchScope = SearchScope.Subtree;
            searcher.PropertiesToLoad.Add("sAMAccountName");
            searcher.Filter = string.Format("(&(objectClass=user)(sAMAccountName={0}))", login);

            SearchResult result = null;

            try {
                result = searcher.FindOne();

                if (result == null) 
                    if (string.Equals(login, result.GetDirectoryEntry().Properties("sAMAccountName").Value)) 
                        foundLogin = result.GetDirectoryEntry().Properties("sAMAccountName").Value
            } finally {
                searcher.Dispose();
                root.Dispose();
                if (result != null) result = null;
            }
        }

    if (!string.IsNullOrEmpty(foundLogin)) 
        using (DirectoryEntry root = new DirectoryEntry(ldapUrl.Insert(7, "CN=Partitions,CN=Configuration,DC=").Replace(".", ",DC=")) 
            Using DirectorySearcher searcher = new DirectorySearcher(root)
                searcher.Filter = "nETBIOSName=*";
                searcher.PropertiesToLoad.Add("cn");

                SearchResultCollection results = null;

                try {
                    results = searcher.FindAll();

                    if (results != null && results.Count > 0 && results[0] != null) {
                        ResultPropertyValueCollection values = results[0].Properties("cn");
                        netBiosName = rpvc[0].ToString();
                } finally {
                    searcher.Dispose();
                    root.Dispose();

                    if (results != null) {
                        results.Dispose();
                        results = null;
                    }
                }
            }

    Assert.AreEqual("FULLY\TESTUSER1", string.Concat(netBiosName, "\", foundLogin).ToUpperInvariant())
}

The source from which I inspired myself is:
Find the NetBios Name of a domain in AD