How do I connect and proof, if an user is in active directory and the authorization is correct.

// create a "principal context" - e.g. your domain (could be machine, too)
using(PrincipalContext pc = new PrincipalContext(ContextType.Domain, "YOURDOMAIN"))
{
    // validate the credentials
    bool isValid = pc.ValidateCredentials("myuser", "mypassword");
}

public List<GroupPrincipal> GetGroups(string userName)
{
   List<GroupPrincipal> result = new List<GroupPrincipal>();

   // establish domain context
   PrincipalContext yourDomain = new PrincipalContext(ContextType.Domain);

   // find your user
   UserPrincipal user = UserPrincipal.FindByIdentity(yourDomain, userName);

   // if found - grab its groups
   if(user != null)
   {
      PrincipalSearchResult<Principal> groups = user.GetAuthorizationGroups();

      // iterate over all groups
      foreach(Principal p in groups)
      {
         // make sure to add only group principals
         if(p is GroupPrincipal)
         {
             result.Add((GroupPrincipal)p);
         }
      }
   }

   return result;
}