What visibility and cloud clearances are required for airplanes conducting visual approaches?
This depends:

Under part 91, you just have to remain clear of clouds and have a minimum of a 1,000 ft. ceiling and 3 SM visibility.

Commercial operations (121 and 135) are further restricted by their Operations Specifications and have to maintain VFR weather minimums appropriate for the airspace that they are in (among other things, see below).

Here is an excerpt from the AIM (there is more details there for different operations, with the differences between controlled and uncontrolled fields, etc.)


  5−4−23(a). Visual Approach
  
  a. A visual approach is conducted on an IFR flight plan and authorizes a pilot to proceed visually and clear of clouds to the
  airport. The pilot must have either the airport or the preceding
  identified aircraft in sight. This approach must be authorized and
  controlled by the appropriate air traffic control facility. Reported
  weather at the airport must have a ceiling at or above 1,000 feet and
  visibility 3 miles or greater. ATC may authorize this type approach
  when it will be operationally beneficial. Visual approaches are an IFR
  procedure conducted under IFR in visual meteorological conditions.
  Cloud clearance requirements of 14 CFR Section 91.155 are not
  applicable, unless required by operation specification.


The Part 135 Ops Spec for this contains the following requirements for a visual approach:


  b. Terminal arrival IFR - Visual approach or a Charted Visual Flight
  Procedure (CVFP).
  The flightcrew may accept a visual approach or a
  CVFP provided all the following conditions exist. The flightcrew may
  not accept a visual approach or a CVFP unless the limitations and
  provisions of subparagraph f. of this operations specification are
  met. 
  
  (1) The flight is operated and remains in Class B, C, or D
  airspace, within 35 miles of the destination airport in Class E
  airspace, or the airspace beneath the designated transition area. 
  
  (2)
  The flight is under the control of an Air Traffic Control (ATC)
  facility. 
  
  (3) The flightcrew must maintain the basic cloud clearance
  as specified in Section 91.155. 
  
  (4) For a visual approach without a
  CVFP - The flightcrew must be able to establish and maintain visual
  contact with the airport or maintain visual contact with the traffic
  to be followed as directed by ATC. In addition, the following
  provisions and weather conditions at the airport during the approach
  must be met: 
  
  (a) Reported visibility must be as specified in Section
  91.155, but not lower than a visibility of three miles and reported ceiling must be 1,000 feet or greater, or  
  
  (b) When in the terminal
  area with the reported visibility not lower than three miles and
  ceiling not reported, the flightcrew may continue to a landing if the
  runway of intended landing is in sight and the flightcrew can maintain
  visual contact with the runway throughout the approach and landing,
  and 
  
  (c) Ceiling and cloud clearance must be as such to allow the
  flightcrew to maintain the minimum altitudes prescribed in Section
  91.129, 91.130, or 91.131, as applicable for the airspace class in which the flight is operated. 
  
  (5) For a CVFP - The flightcrew must be
  able to establish and maintain visual contact with the airport or the
  charted visual landmark(s) for the CVFP throughout the approach and
  landing. In addition, the weather conditions at the airport at the
  time of the approach must be reported to be at or above the weather
  minima established for the CVFP, but never lower than the VFR landing
  weather minima stated in Section 135.205 in uncontrolled airspace.
  
  ...
  
  f. Special Limitations and Provisions for Visual Flight Rules. All VFR operations authorized by this operations specification shall be
  conducted in accordance with the following limitations and provisions.
  
  (1) The certificate holder must identify obstacles and use airport
  obstacle data which ensures that the performance requirements of Part
  135 are met.
  
  (2) The weather conditions must allow the flightcrew sufficient
  visibility to identify and avoid obstacles and safely maneuver using
  external visual references and to maintain minimum altitudes.