The indication "C2 10uF" almost certainly refers to a 10 microfarad (uF) ceramic capacitor. The linked documents didn't include a bill of materials, but the bill of materials would generally be the place to look for details on the parts that should be assembled onto the board.
Also, the positioning of the capacitor is rather weird; IMO the designer should have placed the cap closer to the power supply pin it's routed to. Between this and other inconsistencies (as well as the lack of a bill of materials), I'm not very confident that the project's author paid enough attention to a quality circuit board design. However, with the boards already fabricated, there's not much you can do about that, and this design will probably work well enough anyway.
I ended up downloading the PCB files and measuring the pads of the capacitor using the gerber viewer tool bundled in my KiCAD installation - the dimensions of the pad seem to line up pretty closely with the standard "0805" surface-mount capacitor package.
Finally, the design seems to be using 5 V throughout for power meaning the capacitor needs to be rated to withstand 5 V; putting everything together, this means that you'll need a capacitor that has the following specs:
- 10 microfarad (uF) capacitance
- 0805 package (0603 is probably "close enough" but smaller and harder to solder)
- Voltage rating of at least 5 V, but preferably more for a bit of breathing room. 6.3 V, or even 10 V, is a decent choice.
In this case, you did manage to find hints regarding C2 from the video. However, if you didn't have any such clues, you could trace the connections from the pads and find that they connect to the power supply line (at the barrel jack and Arduino VIN) and ground pour; this suggests that the part is a capacitor (since it's placed across the power supply near a chip, and capacitors are used in such a way to provide power supply decoupling).
Anyway, while I have this board up in a viewer, a few improvements that you could make if you designed your own board:
- Tying 5V to arduino Vin will cause the Arduino 5V supply to sag below 5 V, because of dropout in the arduino's voltage regulator
- All of the ICs should have similar capacitors across their supply lines, close to their supply pins
- The board design looks careless overall, and could be made neater with careful hand layout.
- There are no reference designators or corresponding bill of materials. If I needed to distinguish between multiple ULN2003N ICs, I'd rather be able to say U1/U2/U3, rather than "the middle one".
- The current design has very few markings regarding polarity or pins for the barrel jack and motor connectors
