Yes, you will have to write an application. Personally, I use Delphi5/6 that have socket objects in their library so I just write a simple app that copies data from the socket to a file. As for the format of the data, there's no reason why the 8051 side can't output CSV strings. You could also use VB,C#,C++,Java,PhP etc to write the application as these languages also have socket support.

Recently I've been using the ENC28j60 device in a configuration much like the Tuxgraphics unit and modified their code. Whilst it was for an AVR, to port the code to an 8051 is not much effort - apart from the SPI and some port bits, the code is non-processor specific and it doesn't use interrupts.

Want wireless? Just connect it to a cheap wireless access point - I resurrected my bricked Linksys WRT54 and loaded some open-source software onto it. With an addition of a max3232 chip, you also get a serial port. Instant serial->wireless ethernet.