Hi ociredlu2
It depends what you like to do. If you only want to monitor an input or set some outputs via the internet, this can be done with a simple microcontroller running a TCP/IP stack (look at the microchip's homepage, there are also many others). The ethernet port of such a device has to be connected to the open internet and you can set/read an IO from any location.
If you want to do more complex things, you have to choose a more powerful controller (e.g. a PowerPC MPC8xx from motorola) and a stable operating system with TCP/IP stack (e.g. linux). You can run a webserver or whatever you want.
Bye