You can indeed take the chip out of the board and use it in your own circuit. This diagram maps legs of the chip to pins of the Arduino: http://www.arduino.cc/en/Main/PinMappingSerial functionality is built into the hardware of the ATmega8 chip. What you will lose is the conversion from TTL serial (e.g. 0 and 5 volt serial, like the ATmega8 outputs) to either USB (which happens via the FTDI chip on the Arduino board) or to standard RS-232 serial (which operates at plus and minus 6 to 15 volts; 12 is common, I believe). If you need, however, to talk to another chip over serial, the built-in TTL serial may be exactly what you need. Or, you can include components in your circuit for conversion to USB or RS-232 serial.Other things you'll lose (or need to supply yourself) include: power regulator (to go from the supplied voltage to the 5 volts needed for the board), a 16 MHz clock (without a clock, the ATmega8 can only operate at 8 MHz, and you'll need to change its fuses with a hardware programmer to tell it to do so), the reset button, power LED, etc. But really, there's not too much on the Arduino board, so you'll keep most of the functionality. Whatever LCDs you can drive from the Arduino board you'll be able to drive from a standalone ATmega8.
The ATmega8 in the Arduino gets configured by us (through internal hardware fuses) for an external clock, meaning that it will not run as-is without a clock. If you want to use it without a clock, [...]