I'll have a start at an answer, but not sure I'm able to completely answer it.
In many cases, radicals on the left side of the kanji indicate the "class" or meaning that the kanji belongs to. This seems to be moreso the case with physical objects rather than abstract concepts.
For example, 人 - person - (called "nin-ben" as the left radical): 休、体、代、伝 ([human] rest, body, generation/change, transfer). (At least one) root meaning of each of these kanji involves people (伝える has a very strong sense of "transfer information from person to person").
A better example is something like 魚 (sakana-hen). I can't think of a SINGLE kanji that has 魚 on the left side that doesn't relate to fish/ocean (look at a sushi menu).
Things in "nature" are pretty consistent: the names of trees pretty much always have ki-hen, 木. 松、柳、杉、楓、樫. (pine, willow, cedar, maple, oak). 林 means a grove of trees. From there it gets more abstract - a desk was probably traditionally made from wood, so つくえ is 机.
Once we get abstract, though, it gets messy. 心 means heart, so usually risshin-ben (when it's a line with two marks on the left) is related to emotional/feeling kanji: 慎 (prudent)、慣 (become accustomed to)、忙 (busy). Sure, we can all make the argument that one is "busy" and that affects the heart/emotion, but really, when we start making parallels like this, I start sounding more like stories from "Remembering the Kanji" as opposed to a real history of the character.
NOW, as for radicals on the right side of the kanji, they often provide the sound of the on-yomi prononciation. The kanji 官 means "official", as in like a bureaucrat (in the word 官僚). Yet, the 官 structure is part of other kanji - and takes the onyomi along with it.
The word ryokan (旅館 - Japanese inn)'s "kan" kanji includes it, as does kanri (管理, control, administer). Note that in the case of 管理, it's not on the right - it's below the radical, 竹, called take-kanmuri (kanmuri means "crown", note that it's on top, not on the left). 棺 (ひつぎ - coffin) is pronounced "kan" in onyomi (also note that it has ki-hen, because coffins are often made of wood!).
Hope this helps...