I'm not much of a fan of NeverWinter Nights, so I can only offer the explanation of 'Armor Check Penalty' as it applies to the official Dungeons and Dragon 3rd Edition ruleset, I'd be shocked if this didnt precisely hold true for NWN also however.
Essentially Armor Check Penalty is what accounts for the added difficult of performing some skills depending upon what armor your wearing. All armor heavier then leather armor carries with it a penalty.
For example if your trying to perform the 'Move Silently' skill, it is MUCH more diffiuclt to accomplish if your wearing noisy bulky armor like full plate mail as if you were wearing merely padded armor.
The Armor Check Penalty of Full Plate Mail (-6) would be applied to your die roll for your move silently check, or any other skill that is in any way impacted by armor.
In addition, beyond the regular penalties that anyone wearing armor suffers- If you are not proficient with the armor your wearing then you suffer the armor check penalty on all attack rolls as well as all skill rolls that involve movement.
Armor Check Penalties also have other side-effects, such as if you sleep in a suit of armor with a check penalty of (-5) or worse then you automatically suffer the effects of fatigue the next day.
I can explain 'Offhand Weapon' if you wish, but GuybrushThreepwood has already given you the essential gist of what it is.
Keep in mind my explanation is as per the official 3E ruleset, NWN implementation may vary slightly though I'd imagine it's identical in all major aspects.