How to Identify the primary key duplication from a SQL Server 2008 error code?

If you catch SqlException then see its number, the number 2627 would mean violation of unique constraint (including primary key).

try
{
    // insertion code
}
catch (SqlException ex)
{
    if (ex.Number == 2627)
    {
        //Violation of primary key. Handle Exception
    }
    else throw;
}

MSSQL_ENG002627

This is a general error that can be raised regardless of whether a database is replicated. In replicated databases, the error is typically raised because primary keys have not been managed appropriately across the topology.

This is an old thread but I guess it's worth noting that since C#6 you can:

try
{
    await command.ExecuteNonQueryAsync(cancellation);
}
catch (SqlException ex) when (ex.Number == 2627)
{
    // Handle unique key violation
}

And with C#7 and a wrapping exception (like Entity Framework Core):

try
{
    await _context.SaveChangesAsync(cancellation);
}
catch (DbUpdateException ex) 
   when ((ex.InnerException as SqlException)?.Number == 2627)
{
    // Handle unique key violation
}

The biggest advantage of this approach in comparison with the accepted answer is:

In case the error number is not equal to 2627 and hence, it's not a unique key violation, the exception is not caught.

Without the exception filter (when) you'd better remember re-throwing that exception in case you can't handle it. And ideally not to forget to use ExceptionDispatchInfo so that the original stack is not lost.