How can I get the field names of a database table?

Use IDataReader.GetSchemaTable()

Here's an actual example that accesses the table schema and prints it plain and in XML (just to see what information you get):

class AccessTableSchemaTest
{
    public static DbConnection GetConnection()
    {
        return new OleDbConnection("Provider=Microsoft.Jet.OLEDB.4.0;Data Source=..\\Test.mdb");
    }

    static void Main(string[] args)
    {
        using (DbConnection conn = GetConnection())
        {
            conn.Open();

            DbCommand command = conn.CreateCommand();
            // (1) we're not interested in any data
            command.CommandText = "select * from Test where 1 = 0";
            command.CommandType = CommandType.Text;

            DbDataReader reader = command.ExecuteReader();
            // (2) get the schema of the result set
            DataTable schemaTable = reader.GetSchemaTable();

            conn.Close();
        }

        PrintSchemaPlain(schemaTable);

        Console.WriteLine(new string('-', 80));

        PrintSchemaAsXml(schemaTable);

        Console.Read();
    }

    private static void PrintSchemaPlain(DataTable schemaTable)
    {
        foreach (DataRow row in schemaTable.Rows)
        {
            Console.WriteLine("{0}, {1}, {2}",
                row.Field<string>("ColumnName"),
                row.Field<Type>("DataType"),
                row.Field<int>("ColumnSize"));
        }
    }

    private static void PrintSchemaAsXml(DataTable schemaTable)
    {
        StringWriter stringWriter = new StringWriter();
        schemaTable.WriteXml(stringWriter);
        Console.WriteLine(stringWriter.ToString());
    }
}

Points of interest:

1. Don't return any data by giving a where clause that always evaluates to false. Of course this only applies if you're not

Are you asking how you can get the column names of a table in a Database?

If so it completely depends on the Database Server you are using.

In SQL 2005 you can select from the INFORMATION_SCHEMA.Columns View

SELECT *
FROM INFORMATION_SCHEMA.COLUMNS
WHERE TABLE_NAME = 'MyTable'

IN SQL 2000 you can join SysObjects to SysColumns to get the info

SELECT     
    dbo.sysobjects.name As TableName
    , dbo.syscolumns.name AS FieldName
FROM
    dbo.sysobjects 
    INNER JOIN dbo.syscolumns 
         ON dbo.sysobjects.id = dbo.syscolumns.id
WHERE
    dbo.sysobjects.name = 'MyTable'