public classes and public variables

public class Project
{
    public class ProjectListClass
    {
        public int nPom;
        public string sName;
        public int nTotPom;
        public int nExtraTime;
        public bool bSelected;
        public ProjectListClass(string name)
        {
            sName = name;
        }
    }
    public ProjectListClass lstClass;
    public Project(string name)
    {
        lstClass = new ProjectListClass(name);
    }
}
void button_Click(object sender, EventArgs e)
{
    Project pro = new Project("Hello");
    MessageBox.Show(pro.lstClass.sName);
}

or simpler:

public class Project
{
   public int nPom;
   public string sName;
   public int nTotPom;
   public int nExtraTime;
   public bool bSelected;
   public Project(string name)
   {
    sName = name;
   }
}
void button_Click(object sender, EventArgs e)
{
    Project pro = new Project("Hello");
    MessageBox.Show(pro.sName);
}

or:

public class Project
{
   public int nPom;
   public string sName;
   public int nTotPom;
   public int nExtraTime;
   public bool bSelected;
}
void button_Click(object sender, EventArgs e)
{
    Project pro = new Project();
    pro.sName = "Hello";
    MessageBox.Show(pro.sName);
}

First, fields of the class should be private. For access to the instance from the our-of-class code better use properties.

Secondly, public is only needed if you want to have access from another assembly. In other cases, use internal (or internal protected).

Think about what properties should be read-only. You can initialize them from constructor. It includes your sName.

Do not violate Microsoft naming conventions. Don't abbreviate. Worst thins is your prefixes. Type are all in meta-data, no need to prefix with type. The idea came from C, but C# is rather opposite to C in main ideas.

public class Project
{
    public class ProjectListClass
    {
        public int Pom { get; set; } //give sensible name
        public string Name { get; set; }
        public int TotPom { get; set; } //give sensible name
        public int ExtraTime { get; set; }
        public bool IsSelected { get; set; }
    }
}

//Project Projects = new Project();
//Same name for the variable and class is perfectly fine and often recommended:
Project Project = new Project(); //why it was plural? It wasn't array...

//...

//Here, Project is instance, not class:
Project.Name = MyForm.textBoxName.Text;

Any questions?

—SA