Hello. I'm working on a project of making a Voltimeter. However, I notice when the circuit is not closed, the output jumps all over the place. I was just wondering if it was possible to detect when this happens. Thank you and sorry if I'm forgetting something! 
Add a "pull down" resistor between the analog input and ground.
If you are measuring a power supply or anything high-current/low impedance 1K - 10K is a reasonable value.
"Real multimeters" usually have very-high input impedance (more than 1 meghom) so if you have a high-impedance or low-current voltage source you might want to use 1M or so. (Higher impedance is more susceptible to noise pickup, especially when the input is "open").