Mistakes were made by referees at the 2010 World Cup that could have easily been avoided with the use of technology to assist the ref and his assistants. Here are the areas that soccer can use instant replay effectively.
Goals- If you really think about no official on the pitch is really put in a good position to make decisions about where the ball is in relation to the goal line. The assistant referees are in line to call Offsides and the referee is behind the play. Using replay to check questions regarding the goal line would improve the game, because goals are incredibly significant in soccer and the call should always be made right. Also, if the technology could be perfected a ball that makes noise when it has entirely crossed the goal line would be an excellent option.
Offside- Offside is a really difficult call if you really think about all that has to be taken into account by the assistant referee. To really make an accurate call the assistant referee has to be able to see several areas of the pitch simultaneously. This would be able to be used effectively to disallow goals scored by players who were in an offside position. It becomes more complex correcting the other type of mistake. If a player is was actually onside, but who is called offsides, scores a goal is possible to discern the effect of the offside call on the players on the pitch. I think a good way to determine whether or not a goal should be allowed on an overturned call is to answer two questions. Was the Goal Keeper affected by the offside call? and if so did he have a realistic chance at a save anyway?
Special Circumstances in the Box- There are two things that I would consider "special circumstances." One is handball. An advantageous handball in the box can dramatically alter a game. It shouldn't be allowed to be so, and also should have a player sent off. The other circumstance would be out of bounds calls. This could be corner vs. goal kick or whether or not a ball crossed the touch line before or during a cross.
Overt Aggression- Acts like the kung-fu kick in this year's World Cup final, Zidane's headbutt, or Melo's stomping on Robben are unacceptable in soccer and should see a player sent off the pitch. If acts like this are missed by the officiating crew, replay could make sure acts like those a properly punished.