These two tools allow you to look inside your circuit as it runs. The Volt-Ohm Meter (VOM) is used to debug analog circuits while the Logic Probe is for work with digital ones. None of the circuits we build will require anything fancy from these tools, but here's a bit about them so you'll be sure to get what you need.
The VOM does what it says. It measures voltage, resistance (in ohms), and often current (in amps). Most VOM's have a big dial on the front to select the appropriate range for the level you're measuring. Some VOM's are autoscaling, and choose the range by themselves. Personally, I find that this autoscaling can be annoying, because of the time delay involved while the VOM chooses a range.
You can choose VOM's with a digital display, or with an analog display using a needle. Either are fine for our uses, although the analog display is sometimes better for seeing changes over time.
The logic probe is handy for debugging digitial logic circuits. You connect a clip to the circuit's ground, and then touch the needle probe to different pins in your circuit. The probe will tell you if the pin is being driven high or low, or if it's floating. With the probe, you can trace information all the way through your circuit.
Some VOM's come with a logic probe setting that allow you to use the same box for both purposes. However, it's not a great idea to try and use a normal VOM as a logic probe (just looking at voltage levels), since there's no way to tell if a value is being driven high by the circuit, or just happens to be floating high.