How do I reverse my electric motor?

The components on the back of the motor are three capacitors to suppress the contacts, and a diode to protect whatever switched it from a back-emf.

If you reverse the polarity without removing the diode, then you will just put the current through the diode not the motor, and possibly either destroy it or damage your power supply, or both.

So you need to remove the diode, and if you are using a simple transistor to switch the motor, provide some other protection. If you're using an h-bridge with diodes, then you're ok (I'd imaging the gert-board does, but don't know for certain).

You also need to remove the larger black capacitor - it is polarised, so will be damaged and possibly explode if you connect it the wrong way round. Previously the diode prevented the current going through it when you did that. If the motor generates lots of electrical noise when you run it without the capacitor - interfering with radios and so on - then you will need to add a similar value of non-polarised capacitor to suppress that.

Also check that board you're using can supply the motor's required current.

In addition to Pete's answer (which is completely correct):

You have a permanent magnet DC motor,most likely with carbon brushes.

This would normally be reversible simply by reversing its supply connections, as you first tried (after removing or reversing the diode and big capacitor as Pete recommends.

However it is likely that the motor is tuned to run in one direction only.

This is done in a brushed motor by rotating the brushes in the direction of rotation (a little bit like advancing the ignition on an IC engine) so that the magnetic field builds up in time to do the most good. It also reduces sparking when the brushes break contact. Looking at the motor, there is a slot through which you can observe the brushes : check for visible sparking both ways round. It may also be possible to adjust the brush position through this slot with a suitable tool, if you are brave enough.

However, even if the motor will run backwards well enough, it is driving a ducted fan. The curved blades will stall when reversed, and deliver a fraction of the expected thrust, possibly consuming higher than normal current. Check the current on the bench. The reduced thrust may or may not be a problem in your application...

If you can't do what you need by reversing the motor, mount the whole ducted fan in gimbals and rotate it 180 degrees.

The motor in the question has an electrolytic, polarized capacitor, and a diode, across its leads - this in itself clearly says that the motor was not supposed to be driven with reversed polarity. The capacitor in question is the black can.

The burning smell was either the short-circuit through the diode frying your wiring, or if the diode had already failed, the capacitor getting extremely upset at being connected to an inverted polarity. You may have been fortunate in that electrocaps tend to explosively express unhappiness at being connected the wrong way around. The white strip on the capacitor casing, seen at the left edge of the capacitor in the last photo, indicates the negative voltage connection.

That the motor moved at all in reverse seems to indicate that the diode had already blown, and the power the motor was receiving was whatever was left over after leakage across the reversed capacitor - not much. That may explain the slow reverse rotation.

Do not use that motor even with the correct polarity before replacing that capacitor and the diode. Further, check if the motor can run acceptably in reverse at all, as the other answers have pointed out.

For operating a DC motor in either direction, use a H-bridge circuit:

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Instead of discrete MOSFETs or transistors, you can also use a H-Bridge IC such as the Freescale Semiconductor 5 Ampere H-bridge MC33887 or Texas Instruments DRV8837, so long as the voltage and current requirements of your motor are met by the IC selected.

Also, replace the capacitor with an unpolarized one (for electromagnetic noise suppression, if this is necessary), remove the diode from the motor leads and use 4 suppression diodes at the H-bridge terminals, as shown in the above diagram.

If you cannot find a suitable non-polarized capacitor, see this answer, for a way to achieve unpolarized capacitor behavior from two polarized ones.