How to drive nixie tubes from Arduino
The nixie tube is essentially the same as a neon bulb with multiple cathodes. The wire cathodes are shaped into the numbers 0 to 9. To turn the electrode ON you need to supply a high voltage (of about 170 Volts DC)to the common ANODE (the see through grid). This will be through a current limiting resistor to give a current of about 2mA. When 'lit' the anode voltage will drop to about 140V (the maintaining voltage). You will need to check the actual figures for your chosen nixie tube. 

To light the cathode you will need to switch it to ground (0v) through a transistor or other suitable switching device. 

This could be done by switching each cathode with a single I/O line (through suitable transistor etc.) but with three nixie tubes you would quickly run out of lines. 

This is where the 74141 comes in. It is a BCD decoder 1 of 10 outputs designed to switch the cathodes of the nixie to ground. This requires FOUR I/O lines per digit (nixie tube). This chip also ignores non valid numbers (1010 - 1111) and blanks the display.

You could drive this chip directly from the arduino OR you could add some 4 bit latches between the arduino and the 74141 avoiding the need to tie up the I/O lines or multiplex the display. 

Health and safety warning - High voltage hurts and destroys arduinos if you're not careful.