What is a monitored approach?
A monitored approach is a special kind of instrument approach involving added verbal call outs and increased monitoring of the airplane and is typically conducted when weather is below a certain threshold. For example, a crew may be required to fly a monitored approach if the weather is below 3/4 mile vis and the captain has less than 100 hours in type, or in any case with the weather below ILS Category I minimums. It is also specific to multi-crew aircraft and requires at least 2 pilots and in many cases an autopilot. 

In a crew environment, the pilots are typically designated as Pilot Flying (PF) and Pilot Not Flying (PNF).  In general the PF is flying and making commands of the PNF, and the PNF mans the radios and operates gear and flaps the command of the PF.  More detail of this setup probably deserves its own question.

To illustrate a monitored approach, I'll describe on based on my companies procedures.  The specific requirements can vary between operators based on crew training, aircraft equipment and approved opspecs.

The monitored approach briefing

We have identified the arrival airport current weather is 1200 RVR and obscured ceilings and we plan to fly the ILS to category II minimums.  Myself and my first officer have both have current category 2 training, neither of us are high-minimums, and the aircraft has no MEL items effecting category 2 operations.  We are legal to fly the approach, but company operations require we fly it as a monitored approach.

Regardless of who has been PF and PNF up to this point, the first officer becomes the PF and the captain the PNF.  The PF briefs the approach as normal except for:


CAT II radar altitude minimums
Low vis taxi routes if applicable
The approach will be flown by the autopilot (company specific)
Special autopilot considerations
Required verbal call outs (see below)


The monitored approach

For my company, the autopilot had to be engaged in "CAT 2" mode before glideslope intercept.  This will vary by aircraft and company.  For the approach, as noted earlier, the first officer is PF, captain is PNF.

The PF duties are to monitor the autopilot and to execute the missed approach if the autopilot has problems.  The PNF also monitors the autopilot.  The pilots are monitoring the autopilot for performance and for indications it cannot meet required tolerances for the approach.  

PF will call for gear and flaps as normal, except a non standard flap setting may be used for the approach (e.g. we flew flaps 22 instead of 45).

Once configured for landing, the PF will call for "Landing checklist - to the line" (company specific), this for us was the normal landing checklist with the caveat that the checklist is complete when the captain announces "Landing, I have the airplane" (more on this in a moment).

Nearing the Cat II decision height (typically 100 ft RA), the PNF will announce "Going heads up", discontinue monitoring the autopilot and look for the approach lighting system and the runway lights.

The PF will continue to monitor the autopilot.

The airplane will announce "Minimums" at the radar altitude DH and when the airplane reaches this height, the PNF will fly the missed approach.

If the captain sees the required lighting to continue the approach, they will announce "Landing, I have the airplane", and the FO will reply "you have the airplane, landing checklist complete.

The Captain must disconnect the autopilot (by 50 feet, no autoland for us), and would fly any aborted landing executed after taking control.

Caveats

Rules on autopilot usage, disconnect heights and autolanding will vary based on aircraft and company policies.  My example assumes an Embraer-145 certified for Cat 2 approaches and no autothrottles or autoland capability.  Checklist verbiage and flap configurations mentioned also would vary by aircraft and company.  And finally, the specific callouts will vary by company.

While it seems like there was a lot of Category II specific things in my example, the two concepts of reduced visibility ILS approaches and monitored approaches are intertwined, and you will almost always be flying a monitored approach because it is required to conduct the low weather approach by company policy.