I'm studying up stuff I don't do much in order to prepare for my training and proficiency check. I don't think I've ever done a contact approach. It can be a dangerous maneuver.
An aircraft on an IFR flight plan can follow instrument indications almost all the way to the runway (a precision approach) or to the immediate vicinity of the airport, descending further only if the runway is in sight (a non-precision approach). It is possible also to remain on an IFR flight plan yet stop navigating by instruments and start navigating by looking out the window, before reaching the airport.
If the weather is good -- at least 3 sm visibility, and ceiling at least 500 feet above the minimum IFR altitude -- and the pilot sees both the airport and any traffic which she is supposed to be following, ATC can clear her for a visual approach. She finds the airport, flies to it and lands. She is respnsible for wake turbulence separation, noise abatement procedures, and looking out for VFR traffic. There is no published missed for a visual. If weather might bring on the need to miss the approach then the pilot should not accept a visual approach. A visual is easy. I do those a lot. The weather limits are equivalent to those required for VFR flight in controlled airspace.
If the weather is lousy, and the pilot can't see the airport, but thinks she can find it, she can ask for a contact approach. The visibility has to be at least one nautical mile, the very minimum allowed for VFR flight. The aircraft must be flown at least 1000 feet above the nearest obstacle within five nautical miles of where the pilot thinks she is. The airport must have a published instrument approach. Pilots are cautioned to be familiar with the local terrain and noise abatement when attempting this kind of approach. I think some companies forbid them.
I've just made that sound really sketchy, but there are times when a contact approach can be safe and useful. Your airport is on the shore, and there's always a bunch of cloud right around the MDA. You are at the MEA, but you can't see the airport. You request descent to the MOCA, but you still can't see the airport, although you can see the shoreline and recognize the geography telling you you're almost there, and the AWOS says you have at least a mile in mist at the field. You ask for a contact approach, descend out of the MEA and follow the shoreline until you have the airport in sight and then you turn and land. I guess the contact approach is meant to fill in the gap between cancelling IFR when there's a chance the flight can't be completed VFR, and turning away from an airport where you could land perfectly well if it weren't for the stupid rules.
And I've probably used this joke before, but it fits.
"Centre, Barnburner One, request the visual approach."
"Barnburner One, confirm you have the airport in sight?"
"Uh, negative Centre, but we know where it is!"