Hi, Lexi,
You might not know it, but you've brought up an enormous topic and one that is fundamental to the LSAT, particularly on the Logical Reasoning and the Analytical Reasoning sections. Since an understanding of contraposition requires an understanding of the fundamentals of conditional propositions, I will cite some excellent resources we have at your disposal. At PowerScore, conditionals (and contrapositives) are among the first topics introduced in our LSAT classes. My colleague Jon Denning has compiled the following list of existing material we have written about conditional reasoning both within this forum and on our highly-recommended LSAT blog:
• An amazing, and amazingly extensive, discussion of conditional reasoning diagramming by Dave Killoran, author of the PowerScore Bibles:
http://forum.powerscore.com/lsat/viewto ... =12&t=7610
• Another analysis of conditional diagramming by Dave:
http://blog.powerscore.com/lsat/bid/333 ... To-Diagram
• A thoughtful exploration of just how frequently conditional reasoning appears on the LSAT, by PowerScore Developer Nikki Siclunov:
http://blog.powerscore.com/lsat/bid/288 ... Everywhere
• One more from Dave Killoran, this time breaking down the relationship between contrapositives and circular reasoning:
http://blog.powerscore.com/lsat/bid/322 ... rapositive
• An article in our Free Help Area to assist you when dealing with multiple sufficient and necessary conditions:
http://www.powerscore.com/lsat/help/multiple.cfm
With these resources in mind, to answer your specific question about contrapositives, I find it helpful upon encountering a a conditional statement or conditional reasoning to formulate the proposition in "if...then" form. Once one has this "if...then" statement in mind, it is now useful at least to consider what the contrapositive would be and to ask oneself whether this contrapositive might be useful to answer the question or understand a rule in a game.
For example, given "Only computer scientists understand the architecture of personal computers, and only those who understand the architecture of personal computers appreciate the advances in technology made in the last decade," you can translate these statements into:
If understand architecture of personal computers, then computer scientist.
If appreciate the advances in technology made in the last decade, then understand architecture of personal computers
Now I ask myself what the contrapositives would be and whether they are useful. Flip and negate.
If not computer scientist, then don't understand architecture of personal computers.
If don't understand architecture of personal computers, then don't appreciate the advances in technology made in the last decade.
Here the contrapositives are not only useful but also perhaps essential to understanding this stimulus and answering the question.
However, here's another example of a conditional taken from a rule on a game in which all the items must be advertised at least once:
J is not advertised during a given week unless H is advertised during the immediately preceding week.
First, rephrase as "if...then":
If J is advertised, then H is advertised during the preceding week.
Then symbolize:
J 
[H J]
Now consider the contrapositive:
~[H J] ~J
In this particular circumstance, since all the products must be advertised, the contrapositive
~[H J] ~J does not add much to the understanding of this game (J must be advertised).
To recap, the contrapositive is an invaluable tool for understanding both stimuli in Logical Reasoning and rules on games. It is logically equivalent to the proposition from which it is derived. Therefore it is essential at least to pause to consider what the contrapositive of any given conditional would be. Sometimes it will be more useful than other times. If you are going through the trouble of symbolizing the conditionals in a stimulus, you might consider just going ahead and symbolizing their contrapositives. You must understand at a minimum that if you negate a necessary condition, then you have also negated the sufficient condition.
I encourage you to read over the links referenced above. Should you have further questions, please follow up, and I will do my best to address them.
