Is it as simple as just ignoring the letter? Or would I need to pay officials off, or go into hiding?
Well I'll need more details. You might be deferred for various reasons. You might be unfit service. Don't ignore the letter, there's a common misconception that getting the letter means you are going to war. Actually you will just be registered. Remember that the Napoleonic Wars start this idea of mass conscription or "Levée en masse".
Being drafted means you have to register though upon further examination the military may defer you or deem you unfit for service. All sound candidates have their names put in a glass bowl lottery for actual drafting into military service. Let's not get in trouble with the law so let's show up to be registered. You will be given a classification and if any of the following categories apply to you you may be able to avoid the military entirely or at least be given a noncombatant role.
Are you a conscientious objector? Religious reasons can be used. Quakers mad up most of COs. COs maybe put into the Civilian Public Service as mentioned by /u/fallen_seraph or a noncombatant role. You would actually prefer the noncombatant role as the Service is long and hazardous work without compensation.
Do you have minor defects that can be corrected? If they can be corrected by eyeglasses, false teeth, etc. and you are a CO you can be put in noncombatant roles.
Are you a student? You can be a CO and then be out in a noncombatant role. You could also be put to work in alternative service.
What's your occupation? Some people were deemed unfit and left to fulfill an occupation in agriculture, war industry or occupations deemed necessary for nation health or safety or interest.
How well off are you and do have any dependents? People whose absence will cause hardship and privation on their parents, spouse, or children or are already under extreme hardship or privation may be deferred.
How old are you? If you are over 45 you are not going to be serving. This maximum age decrease as the war went on but at the beginning 45 was the upper limit.
Do have a significant physical/mental/moral defect? My grandfather who volunteered but was rejected for having a bad knee. Other physical impairments or disabilities will get you out as well. If you are illiterate you can't serve. If you have a mental disability you can't serve. If the military would be genuinely scared about having you near weapons and other people you obviously won't serve.
Sources: http://www.history.com/this-day-in-history/united-states-imposes-the-draft
If you want the document: http://www.docstoc.com/docs/79930353/Determination-of-Knowing-and-Willful-Failure-to-Register-Military-Selective-Service-Act-(MSSA)
I know Wikipedia is not the best but it had a decent list of the categories: http://en.wikipedia.org/wiki/Selective_Training_and_Service_Act_of_1940