I am using this linear voltage regulator. The datasheet indicates the input and output values for the capacitance to use, 1uF and 10uF respectively.
Should these capacitors be or a particular type, or does it not matter?
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It doesn't usually matter, but be aware that some linear regulators--the popular LM2940 series, for example--may be unstable if the output capacitor's ESR is too high or too low. As the datasheet for your regulator doesn't seem to say anything about that at a glance, it should be fine with any capacitors you pick, but see the edit below for a warning.
Non-polarized capacitors more than about a microfarad used to be rare and expensive, which is probably why the datasheet shows polarized capacitors being used. Today, you can get 10μF ceramic capacitors for less than $0.30 each.
Edit: As @ThePhoton points out, this regulator may be so old that multi-microfarad ceramic capacitors, with their inherent low ESR, may have been a far-off pipe dream to the engineers writing the datasheet. So this may still be unstable with too low an ESR on its output, so unless you want to test its stability under different operating conditions with the ceramic caps, it may be best to stick to aluminum electrolytics. After all, that's probably what the IC's designers had in mind.
The datasheet application circuit example schematic shows a 1 microfarad polarized capacitor on the input and a 10 microfarad polarized capacitor on the output. Since the values are in the 1 plus microfarad range and the capacitors are shown as polarized, I would guess that the manufacturer (ST) wants you to use electrolytic caps. I guess you could use a tantalum caps, but unless the datasheet specifies tantalum, that would be a needless expense.
The polarized caps shown on the datasheet circuit lead me to believe that electrolytic caps are what are intended. Very few ceramic caps are over 1 microfarad and very few are polarized.
In contrary to the previous answers, it does matter.
The electrolytic cap is very good at storing a large amount of energy, but bad for high transient. Having large transients on an electrolytic cap will shorten its lifespan.
Ceramic capacitor store less energy but are good at coping with large transients.
That is why, a good implementation, as shown on the datasheet, is a large electrolytic cap that is there to store the energy, and a ceramic cap, as close as possible to the switching element, to take the switching transients.
If you only put an electrolytic cap, the lifetime of your circuit will be impacted, and only a ceramic cap, your circuit might be susceptible to issue when you have rapid power draws or input line unstable feed.