Can I measure the current (mA) of my project with just a multi meter? I have a project and wish to choose the correct battery back for it such that it will last the required amount of time between each recharge of the battery pack. I am using an Arduino Duemilanove with various attachements and shields.

I can measure that the draw on the project at the point the USB connects from my PC to the Arduino board (by attaching the multimeter to the Vcc and GND solder points on my Arduino board) which reads 5V DC as expected. Is there a way I can meausre the current (without any additional equipment) ?

I suspect somehow using Ohms law and measuring the resistance between two points would be the answer (knowing the voltage is 5V) but I can't work it out (as in where to measure) with my lack of any electrical knowledge.
No, you won't get the right result measuring resistance. Integrated circuits contain lots of semiconductors which are inherently non-linear. 

You need to somehow put your multimeter in series with the power supply, set to the mA range. The easiest way I can think of is to use an external battery (7v or higher) and make up a custom cable with standard barrel jacks. As your board appears to have a linear voltage regulator, the current measured will be very close to the current consumed when powered from the USB port.