Charging devices - Voltage and Amperage

Note: Main OK point from below is that a power supply with the correct voltage rating and a higher than specified current rating will be OK to use in the vast majority of cases. The situations where this does not work are so rare as to be worth risking unless major safety or $ value concerns exist. Even then you can probably check with over current supply at correct voltage and see if problems occur. If the equipment is worth over 2 weeks income get the correct supply :-).

Power = Volts x Amps = V x I

Too high Voltage will do damage.
Equipment may be damaged Equipment MAY shut down Slightly too high may be OK. YMMV.

Too low voltage may do damage. Slightly too low may work AOK or at reduced current. Too low may damage equipment but not usually.
Too low may damage charger but not usually.

Correct voltage rating and too high current rating is USUALLY OK - equipment will take what it needs.
A very few items of equipment will object in some way.

Correct voltage rating and too low current rating may cause damage.
Slightly too low may be OK with computers an similar.
Older equipment using power supply as charger will quite often work with current rating far too low and just charge slower. Too low current rating - equipment MAY load power supply down - MAY damage supply - MAY work - MAY just refuse to work. eg Most notebooks / laptops / tablets - typically with 15 to 20 Volt supplies - will not even try to charge if power supply voltage is more than a Volt or two too low

Using an adapter with under-rated current could result in a couple of different outcomes, depending on the particular adapter. (Assuming that since you said 'device' and not just 'battery', that we're talking about devices that have managed batteries in them, which is pretty common now)

A very cheap one might overload and burn out. Device will take very much longer to charge.

A better one, with protection will reduce its output voltage to keep the current constant, in which case, due to the reduced voltage, your device won't charge much faster than in the first case.

You actually want an adapter that has as much, or more, current capacity than the device you're trying to power. The current marked on the adapter isn't the current you necessarily get, you see, it's the upper limit of what the adapter wants to provide. The amount of current you actually get depends on the device, and if it demands too much, your adapter will suffer.