Reading values through Data Reader

You need to loop through all the rows on your data table if you want to compare them to your textbox. You can try something like the following:

foreach(DataRow row in dt.Rows)
{
        if (txtUser_Name.Text == row["User_Name"].ToString())
        {
            lblUnavailableUser_Name.Visible = true;
            lblAvailable.Visible = false;
        }
        else
        {
            lblAvailable.Visible = true;
            lblUnavailableUser_Name.Visible = false;
        }
}

Sorry to say but your code is messed up. Reasons:
1. Why execute the command in different event and load the values in different event?
2. What are you trying to achieve with this line:
if (txtUser_Name.Text == dt.Columns["User_Name"].ToString()).
This will just give the expression of the column (not value).

If you are trying to check if the entered user name exists in DB or not, apply filter to your query and make it something like this:
"select User_Name from Registration where User_Name = " + txtUser_Name.Text
and use ExecuteNonQuery method in place of "ExecuteReader" as you just want to check for the existence of a value.

Hope that helps!

You should make a loop, on the datatable rows e.g.:

foreach(DataRow row in dt.Rows)
{
  if (txtUser_Name.Text == row["User_Name"].ToString())
  {
    //...
  }
  else
  {
    //...
  }
}

BTW probably the ToString() call isn't needed.

I think it will help you.. I've improved my solution so it is more useful to you

public partial class AJAXTestPage : System.Web.UI.Page
{
    SqlConnection con;
    SqlCommand cmd;
    SqlDataReader reader;
    DataTable dt = new DataTable();
    protected void Page_Load(object sender, EventArgs e)
    {
        if(!IsPostBack)
        {
           con = new SqlConnection(ConfigurationManager.ConnectionStrings["conn"].ConnectionString);
           con.Open();
           cmd = new SqlCommand("select User_Name from Registration", con);
           reader = cmd.ExecuteReader();
           // do another thing with the reader which is needed first time.
           con.Close();
        }
    }
    protected void btnCheck_Availablity_Click(object sender, EventArgs e)
    {
        con = new SqlConnection(ConfigurationManager.ConnectionStrings["conn"].ConnectionString);
        con.Open();
        // here i'm considering userName is a column name of your table, so change the name as per need...
        cmd = new SqlCommand("select User_Name from Registration where userName = @userName", con);
        cmd.Parameter.AddWithValue("@userName",txtUser_Name.Text");
        con.Open();
        reader = cmd.ExecuteReader();
        if (reader.HasRows)
        {
            lblUnavailableUser_Name.Visible = true;
            lblAvailable.Visible = false;
        }
        else
        {
            lblAvailable.Visible = true;
            lblUnavailableUser_Name.Visible = false;
        }
        con.Close();
    }
}