Well, first of all, i'm not an american, but the country i live in does not require service in the military. It did until a few years ago, but fortunately for me that changed.
As you can probably guess from that, i am against compulsory military service. Why? Oh man, so many reasons, where do i even start? In no particular order(Some of these were already mentioned by other people.):
It wastes money. Training people that don't want to be there in the first place will never give results as good as training people that volunteered.
It's not necessary. Todays wars are not fought like the wars of the past century. The days of conscript armies and masses of infantry have come and gone. Sending large numbers of conscripts agains professional soldiers will just end up in a massacre.
Therefore another reason: conscripts will never be able to stand up to professional soldiers.
If the situation is so desperate that a country needs to conscript it's civilian population in order to defend itself, then that country has probably just lost the war anyway, and all this will acomplish will be more waste of life.
It would force some people into doing something that stands against their moral standards and convictions. Lets face in the army, at one point or another, you are tought to kill. Some people have strong feelings about that, for whatever reason. (Myself being one of them.) I don't think it's ever justified to force people to learn how to take another human life.
Some people just aren't "cut out" for army life. I honestly don't think everyone can past even basic training. And i don't mean people that have serious health problems either. I actually think that there are regular people with no particular dissabilities, that can't cut it. To those people it would just be a period of "torture" before being dropped out since they can't take it. I don't see the point in making someones life hell for a period of time, for no gain in the end.
If someone is recruited against his will, then that person will be virtually robbed of a part of his life. There are people that just want to go to university, or get a job... you know, just LIVE their lives, not be stuck in some crappy army for a year or two. And maybe those people don't care about what the army has to teach (courage, valor, discipline, blablabla). So to them it's just a period of life wasted. Is someone going to give them that year of their lives back? Don't think so.
I think i'm just going to stop here.
Oh yeah, and also: "but i dont wanna go!"
I think it's bad idea and in no way would i want to be drafted. If they tried, i'd just dodge it however i could.