When I hit my Irons I get a shock (like hitting a baseball with the thin part of the bat). What am I doing that is causing this? Is it because I am hitting a certain part of the clubface? The shots I hit generally travel straight and have a normal trajectory, with no slices.
Like Donzi said, you are not hitting the sweet spot. You could be hitting off the toe, or of the bottom grooves of the club. See if you can find a way to find out where you are making contact with the ball. If you can't tell from just looking at the clubface, use some tricks, like putting some talcum powder on the clubface (or some purpose-designed impact-powder), or some masking tape (or a purpose-designed impact sticker), or mark up the impact area of the ball with a felt-tip pen, which will in turn leave a mark on the clubface.
The vibration (as others have said) is from not hitting the center of the club face but keep inmind that some iron heads will offer alot more feedback over others. Thicker toplines will not vibrate as much as thin toplines, type of steel and how they are made, graphite versus steel shafts... etc
You will want to make sure you are hitting the center on most cases but when it comes to confidence.... vibration can burst it.
If it takes away from you enjoying the game, make some changes. You might want to look at putting sensicore stripes in your shaft to help elimanate the vibration feeling. It doesn't cost that much and if you can't find anyone locally that does it I would be glad to assist.
I have been in the game for a long time and can tell you that a major part of golf enjoyment is FEEL.
To alot of players FEEL = CONFIDENCE