Your Arduino output ports can't supply the kind of current needed by your motors. Worse, they are not prepared to take the inductive spikes that can be caused by brushed DC motors.
The solution to both problems is to use either a relay, or an H-bridge (with kickback diodes). Unfortuantely most H-bridges are not optimized for 6V, most prefer 12V or higher. For low currents you might check an LD293D, although it won't do that well at 6V.
If you use a relay note that you can't connect a relay directly to the Arduino pins. There are plenty of relay boards for Arduino that combine a relay with a transistor and the kickback diode (which is now needed for the relay coil).
Note that it might not be a good idea to use the same power for both the motors and the Arduino. Start with two separate battery sets. If you get everything working you might try combining the batteries, maybe with some spike suppression. If anything goes wrong then you know the likely cause.
Given this definition, does this mean that every electric component that has electrodes (i.e. anode or cathode), is a nonmetallic part?
Under that definition, in order to have electrodes a component must have nonmetallic parts. But it can't be entirely non-metallic since the electrodes are part of the component and they are metal. For example, a vacuum tube is generally made from metal, glass, and empty space (vacuum). It has both metallic and non-metallic components.
And what exactly is the use of an electrode? Is it just used to indicate polarity?
The electrodes are typically the path for current to flow in and out of the component. The exact metal used might have an effect on the performance of the part, for example in a Schottky diode.
The name of the electrode depends on the polarity: the electrode where current flows in to the component is called the anode and the electrode where current flows out is called the cathode. In the case of zener diodes, these definitions are somewhat abused.
Are there other types besides anode or cathode?
A device with more than two electrodes (for example a transistor or a triode, tetrode, or pentode vacuum tube) necessarily has electrodes that aren't called the anode or cathode.
In vacuum tubes, there's typically an anode, a cathode, and one or more grid electrodes.
In a transistor there is no anode or cathode, just base, emitter, and collector for BJTs, or gate, drain, and source for MOSFETs.
Can I use electrode as a synonym for pole?
I'm not aware of any case where that would make sense. We normally talk about magnets having poles, and electronic devices having electrodes, terminals, pins, pads, contacts, leads, etc.
if the electrode is dependent on the path for current to flow in and out of the component, then does that mean the cathode could switch terminals of a component if the current is reversed?
In principle, this is true. In practice, we choose one terminal of a device to call the cathode and one to call the anode, based on the "normal" use conditions, and we don't change the names when the current direction changes. In the case of zener diodes, we even name the cathode and anode according to what they would be if the part were a rectifier diode, but we normally use a zener with current flowing in to the terminal we call the "cathode".
Best Answer
The anode is where current flows into the device (that is electrons flow out of device); the cathode is where current leaves the device (that is electrons flow into the device). You can use the ACID mnemonic to help you remember (Anode: Current Into Device).
I think there are two possible sources of confusion:
For example when you say electrons flow from the anode, do you mean that they flow into the circuit and out of the device (i.e. cell)? If so you are correct. Note the usage of the terms is consistent in CRTs, within the actual tube (the device) electrons flow out of the cathode, but from the perspective of the circuit driving the tube, the electrons are going into the cathode.
Note that which terminal is the anode and which is the cathode is defined by the direction of current flow not polarity, so in power sources (e.g. discharging batteries) the positive terminal is the cathode, but in power sinks (e.g. charging batteries) the positive terminal is the anode.