In the case of the 7805, which is unconditionally stable, the type and value of the capacitors is not very important- as the note says, you can leave them out entirely and it will 'work' and will not oscillate. It will work better with the minimum capacitances shown and will be happy with much bigger caps (47uF on the input and 1uF-10uF on the output- especially ceramic caps on the output- is always good with 78xx). However, assuming this is a bad habit to get into as most modern regulators are not so forgiving.
The type is not as important as the value, the voltage rating and the ESR. Generally, lower ESR is good, but some older LDO (low drop out) regulators don't tolerate capacitors that are too bad or too good all that well, and can oscillate. Use a value that is too low and they can oscillate. Use a voltage rating that is too low and they can fail early. Usually there is no penalty (other than cost and size) to use a higher than necessary voltage rating, nor to use a somewhat larger than necessary value of capacitance (often tolerances of electrolytic caps are -20/+80% so they might be bigger than marked anyway).
If you have fast logic and that sort of thing you'll want to use a ceramic capacitor (with its low ESR) or a low impedance electrolytic, possibly in parallel with a lower value ceramic capacitor. The input capacitor ESR isn't usually too important on linear regulators, but lower is always better there.
In some cases you may actually have to degrade a ceramic (output) capacitor by adding a resistor of around an ohm or three in series to keep the regulator happy!
In the circuit you show the regulator will be stable and will work with capacitors from 0 to thousands of uF on the input or the output (a 0.33 electrolytic or 0.1 ceramic or greater on the input is advisable if you're far from the input filter cap). If you've got a microcontroller or logic on the output 0.1uF and/or 1uF caps on the output near the loads are required.
The first time you pick up a new regulator datasheet, read it carefully, and search for "capacitor" and "stability" to ensure you've not missed anything. Look at any related graphs very carefully. For example, the 7905 (negative 5V linear non-LDO regulator) does require a minimum capacitance on the output for stability.