There are three levels of rechargable device, and three levels of charger.

A device or battery can only be charged by a charger of equal or higher level.
A battery can only discharge into a machine of equal or higher level. A machine will not blow up through having the wrong battery fitted.
A level 1 machine's level may be raised by fitting 1 or 2 transformer upgrades, though some machines lack upgrade slots (Miner).

Level 1 batteries:
Re-battery (also has a right-click function)
Batpack (will also recharge tools if equipped as chest armor)
If you have Gregtech then the Lithium battery and Lithium batpack are also level 1 meaning you can put 600,000eu in a standard machine's power slot

Level 1 devices:
Electric Jetpack
Power tools (crafting recipe includes re-battery)

Level 1 chargers:
Batbox
Charging bench Mk1

Level 1 machines
All basic industrialcraft machines
Charging bench Mk1

Level 2 batteries:
Energy Crystal
Lappack (will recharge a laser, saber or to tools if equipped as chest armor)

Level 2 devices (crafting recipe includes an energy crystal):
Mining laser
Nanosuit Armor
Nano Saber

If you have Gravitation Suite then several GS items are level 2 when you'd expect them to be level 3, particularly the Advanced Nanochestplate, which charges in a level 2 slot despite having a lapotron somewhere in its crafting.

Level 2 chargers:
MFE
Charging bench Mk2

Level 2 machines:
Machine with a transformer upgrade
(Note that to my knowledge none of the advanced machines will run from batteries)
Charging bench Mk2
Charging bench Mk1 with an upgrade

Level 3 batteries:
Lapotron Crystal

Level 3 devices (crafting recipe includes a lapotron crystal):
Quantum Armor

Level 3 chargers:
MFSU
Charging bench Mk3

Level 3 machines
Machine with two transformer upgrades
Charging bench Mk3
Charging bench Mk2 with an upgrade
Charging bench Mk1 with two upgrades

Footnote: Blutricity (Blutricity follows its own seperate rules)

Batteries:
bt battery
Devices:
Sonic screwdriver, note this is not rechargeable in ANY charger, it only accepts power from a held bt-battery (or it can be "repaired" with [banned] EE items)
Chargers:
bt-batbox (upper slot)
Machines:
bt-batbox (lower slot) only, no other redpower machine has a battery slot