Which hold items are more effective during the battle phase? I've seen "duplicate" items throughout the game, such as the Zap Plate and the Magnet which both increase electric type moves. But which one gives you a greater increase to damage? I know there are others, such as the Flame Plate and Charcoal to increase fire moves, so how many different variances are there of these items and what does each do?
Zap plate and Magnet are the same boost. Any two items that read with the same effect generally have identical effects; the reason they differ are because they have different effects on different Pokemon, usually one is an evolution item for a specific pokemon. In the case of the elemental Plates, they are actually items for Arceus to hold, they change his form and typing due to his Ability, Multitype. The Plates are probably included because there's so many it wouldn't be practical to give them all out when they eventually distribute Arceus in a gen 6 event.
If you are talking about competitive battling, neither the Zap Plate nor the Magnet are used since they boost the power of moves by too little.
In 5th generation, if you wanted to boost moves in Competitive play, you will most often see the gems, in this case the Electric Gem which boost the move by 1.5 times (or 150%!). Unfortunately, the gems aren't available in 6th generation with the exception of Normal Gem, which only gives a 1.3 times boost.
The downside is that the Gems are a one time use, so it's not really practical to use it ingame as you'll be losing the item fairly quickly.
If you like to 'spam' the same moves in succession, I suggest the Metronome which starts at 10% boost to a maximum of 100% boost, and the item isn't consumed. This isn't used in competitive battling either though.
EDIT: If you want to increase the damage output of Pikachu specifically, you can use a Light Ball, which doubles the Attack and Special Attack of Pikachu :)