How are runway conditions measured?
Measurement

Runway conditions are measured using a friction standard: ICAO Circular 329, Runway Surface Condition Assessment, Measurement and Reporting.

Local regulation applies too, e.g. for UK: RA 3272 – Continuous Friction Measuring Equipment.


  Adequate runway friction characteristics are needed for three distinct
  purposes:
  
  
  deceleration of the aeroplane after landing or a rejected take-off;  
  maintaining directional control during the ground roll on take-off
  or landing, in particular in the presence of cross-wind, asymmetric
  engine power or technical malfunctions;  
  wheel spin-up at touchdown.  
  


(Source -- Source also used for most of the technical details in this post)


Continuous Friction Measuring Device / Equipment (CFMD / CFME)  



(Source: Team Eagle)


(Source)


Decelerometer, Brakemeter, Dynometer... for snow and ice.  



(Source: Sherwin Industries)

Minimum Friction Level (MFL) μ is specified by local air authority. A value less than 0.34 is seen by ICAO as slippery.

Friction is measured for each third of the runway, in a direction parallel to the runway centerline.


(Source)

Reporting

Runway friction coefficient should be measured at least daily and reported via NOTAM if needed. NOTAM codes for friction measurement equipment:


BOW Bowmonk Decelerometer
BRD Brakemeter-Dynometer
ERD Electronic Recording Decelerometer
GRT Griptester
MUM Mark 4 Mu Meter
RFT Runway friction tester
SFH Surface friction tester (high pressure tire)
SFL Surface friction tester (low pressure tire)
SKH Skiddometer (high pressure tire)
SKL Skiddometer (low pressure tire)
TAP Tapley Decelerometer


A contamination terminology is used for a qualitative description of the runway condition. It can be used in complement of the friction coefficient value, or alone when a measure is not possible.

Sample for water on the runway:


DAMP: the surface shows a change of colour due to moisture.
WET: the surface is soaked but there is no standing water.
WATER PATCHES: significant patches of standing water are visible.
FLOODED: extensive standing water is visible.


Other terms exist for snow, ice, etc.


  A runway or portion thereof shall be determined as being slippery when
  wet when the measurements with a CFME + self wetting tyre shows that
  the runway surface friction characteristics are below the minimum
  friction level specified by the State.


NOTAM sample:

NOTAN A1713/11 Chennai Airport ( VOMM ),  India
A1713/11 1111220910/1112231200 EST RWY
RWY 07/25 LIABLE TO BE SLIPPERY WHEN WET.
THE VALUE OF COEFFICIENT OF FRICTION IS BLW 
THE NOTIFIED VALUE OF 0.34 FOR RWY 07/25
IN SECTION AS BLW :-
RWY07 BTN 400M AND 650M
RWY25 BTN 300M AND 500M


As commented by @RonBeyer (thanks!), pilots can also provide their own observations about braking conditions to ATC using PIREPS. PIREPS are voluntary reports by pilots to share information with ATC and other pilots. They are valued because they are current and they may be the only information available at small airfields. 

ATC and recorded bulletins provide runway conditions to pilots, in addition of NOTAM.

Impact on roll distance

Weather has many impact on operations, but runway contamination particularly impacts landings, due to the loss of braking capacity. Take-off are also impacted, mostly at the beginning of the take-off roll, when the control surfaces are not active. Runway condition is also important to take into account if a take-off has to be rejected.

A degraded runway condition decreases the longitudinal stability and increases the roll distance:


Note: As mentioned by @Yos233, autobrake LOW and MEDIUM have been inverted
(Source)

Thrust reversers are particularly useful when the runway is not dry:


Note: Autobrake LOW and MEDIUM still inverted
(Source)

Pilots will use similar tables to determine if they can land or take-off. A flight management computer can also be fed with the runway condition for roll distance determination..

Aircraft are tested for their ability to cope with different runway conditions:


A340 test on artificially flooded runway (source)

Note that it is possible to land and take-off quite easily on compacted snow -- and, albeit more difficult, on ice -- taking into account the coefficient of friction:


Alton Bay runway (source)



More


Description of Various Friction-Measuring Equipment
Used in Winter Operations (appendix A)