A contractor aka consultant is a professional who provides expert advice in a particular domain or area of expertise.
Contractor vs. Employee
Contractors typically use their own supplies to do a job, and when they are paid, the organization doesn't take out any taxes. Contractors are then able to write-off supply expenses on their taxes.
As compared to employees who use the organization's supplies to do the job and when they are paid, the organization does take out taxes.
IRS Tax Forms
Before hiring a contractor have them fill out an IRS Form W-9. This mainly gets their Social Security Number, address, and signature. If you don't do this, and the contractor refuses to sign a W-9 after they have done the work -- the organization, tax-exempt or not, has to pay the taxes.
You need to fill out an IRS Form 1099-MISC before January 31st for each contractor or company that you pay over $600 in the tax year. You do not need to file for under $600.
NOTE: The IRS makes a new 1099-MISC form each year, so make sure you are filling one out for the last full year. So if the date is Jan. 15th, 2008 and you are filling out a 1099-MISC for Jane Doe, you should be filling out a 2007 form not 2008.
NOTE: You can't just print out and send in a 1099-MISC from the web, there is a $50 per 1099-MISC form for doing so. You must pick up copies from your local IRS Form office or order them which takes roughly 10 days. The 1099-MISC is an optional form, so call to make sure they have it.
You then need to mail a single IRS Form 1096 and the red copies of each 1099-MISC to the IRS before February 28th. The IRS refers to it as a cover letter.