The Raspberry Pi and all microcontrollers used in the Arduinos natively support a number of standard protocols, e.g. RS-232 (often called "UART" on embedded platforms), SPI and I²C/TWI, for each of which you will find plenty of tutorials.
RS-232 is probably the most robust of them when it comes to long transmission distances, but at the 10 to 15 meters you're expecting, you should reduce the bit rate to avoid transmission errors.
RS-232 in its simplest form only requires a common ground wire and one data wire per transmission direction.
You will be using RS-232 with TTL levels (0 Volt/+5 Volt instead of the up to -15 Volt/+15 Volt defined by the original standard).
Be aware however that the Raspberry Pi operates at 3.3 Volt, while the Arduino is likely to use 5 Volt — enough to damage the Pi — so you will need a level shifter of some sort. Since you're not going to need (or get) high speed anyway, a resistive voltage divider will be sufficient (there are tutorials for that too).