How does a chiropractor adjust the neck gently?

They don't. Chiropractors do a deep massage called thrusting to presumably correct subluxations, something only they can see on an x-ray. Chiro manipulation of the neck is risky and potentially can cause strokes and paralysis. If your neck is bothering you, see your orthopedic specialist first and get some advice and perhaps gentle physical therapy.
Many techniques. There are a variety of techniques utilized in chiropractic some are by hand, others with an instrument. The key to manual adjustments is speed, timing and short thrusts. While chiropractic is safer than aspirin, if you have concerns, let your chiropractor know. As they adjust necks all day long, they have dealt with someone who is not used to receiving adjustments yet. Try instrument adjustments.