suppose I want to keep an Arduino powered during the night, and recharge it during the day with a solar cell. What kind of circuit do I have to make, how many and which type of batteries, and what size of solar panel do I need to keep it going?
I'd start by measuring the power consumption of the Arduino and figure out how to put it to sleep when possible. From what I've read, the Arduino Pro Mini consumes something like 5 mA at 3.3 V, which is around 15 mW. With a solar panel, your capacity factor is likely to be around 10 percent, meaning that it will produce an average power of around 10% of its peak power. This suggests that you need a panel that can produce 150 mW, but I'd probably go for a 1 W panel unless size is an issue.
For batteries, you want to get as low a voltage as you can, while still being above 3.3 V. I might try a 6 V lantern battery.
For wiring, put all three devices (Arduino, battery, and solar panel) in parallel. Example (minus the battery) here.