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