In your circuit, the only two critical capacitors are those at the output of the LP2950 voltage regulator: C4 and C5.
For all others you can pick whatever you want: I'd probably go with electrolytics for any polarized and cheap X7R ceramics for the rest.
For the two output capacitors of the LP2950 there is an entire chapter in the data-sheet: http://www.ti.com/lit/ds/symlink/lp2951-n.pdf Check chapter 188.8.131.52.1
Long story short: Use a low ESR electrolytic capacitor for C4, rated at at least 10V. C5 must not be a ceramic as explained in the data-sheet so I'd either use a high quality electrolytic capacitor or just leave it out it doesn't seem nessesary at all.
(for example why dus he sometimes use 1µF but in other parts 1000nF,
use 47n instead of 47nF or use 4µ7 instead of 4,7µF)
He's using short form notation and hints. If you have a capacitor labled 47n the Farad unit is implied because it is a capacitor. 4µ7 is shorter to write than 4.7µF. It takes less space in the schematic and on a PCB silkscreen.
You'll also sometimes see capacitors values as .1µ or .1u which is the same as 0.1µ or 100n. Again it takes less space on the schematic.
The 1000nF is the same value as 1µF, but it gives a hint that a ceramic capacitor should be used here because ceramics have for a long time not been available in the µF range. You won't see this kind of hints often though.