What is a SAAAR / RNP AR instrument approach?
First allow me to give a quick introduction to RNAV and RNP before geting to SAAAR / RNP AR.

Area Navigation (RNAV) is a method of instrument flight rules (IFR) navigation that allows an aircraft to fly on any desired path within the coverage of referenced navigation beacons, rather than navigating directly to and from the beacons. In other words, waypoints can be defined at arbitrary locations and paths betweens those points (and traditional navigation beacons as well) can be defined. 

RNAV routes are coded into the navigation database by ARINC 424 path-terminators. Whilst RNAV routes can be designed in such a way that the lateral path is strictly described, in general the lateral path has a quite some freedom due to floating leg types (e.g. turn at altitude, fly heading etc) and wiggle room for the aircraft to negotiate the turns. 

When the route is described only by TF (track to fix) and RF (radius to fix) path terminators, the lateral path is fully fixed. These fixed lateral paths allow procedure designers, for example, to design approaches through mountain valleys down to the runway. 

RNAV defines a 95% accuracy bounds. For example, RNAV 1 requires the aircraft to be within 1 NM from the path centreline 95% of the time. The route designer will ensure a 2 NM obstacle free zone around the path in such a case. 

Note that RNAV provides no assurance against exceeding the 2 NM zone. There is no automated system monitoring required. 

Potentially causes of deviations between the path intended by the procedure designer and the actual flown path can be grouped into three categories:


First, there is the Path Definition Error. This is caused by database
coding mismatch with the original path design.
Second, there is the Path Steering Error, caused by the combination
of display inaccuracies on the Navigation Display and the inability
of the pilot or the autoflight system to stay exactly on path (Flight
Technical Error).
Third, there is Position Estimation Error, which is the difference between the position of the aircraft estimated by the navigation system and the true position.


It is not unthinkable that while normally (95% of the time) the aircraft will fly within e.g. 1 NM of the intended centre line, there are cases that the aircraft will exceed the 1NM by far, possibly even busting the 2 NM obstacle free zone. That could potentially ruin your day if it happens in a mountain valley.

This is where RNP comes in. RNP (Required Navigation Performance) is basically RNAV with added protection. For example, RNP 1 again defines a 1NM accuracy for 95% of the time, but it also offers 99.999% assurance that it will not exceed the 2NM. It will raise an alert if the 2 NM value will potentially be exceeded. So in addition to what RNAV offers, it includes a system monitoring function that will detect navigation errors and alerts for them.

The RNP value alway gives the 95% accuracy, the 99.999% assurance (integrity) limit is always at twice that value. So RNP 0.3 means that the aircraft has to stay within 0.3 NM of the designed path centreline 95% of the time. And if it exceeds the 0.6 NM threshold, an alert will be raised with 99.999% certainty. 
RNP systems therefore require a system monitoring function.  

To ensure that the RNP values are met, each of the three error sources mentioned earlier must be under control. For RNP values down to 1 NM that is not much of a challenge. Standard procedures and equipment are deemed to be sufficient here. (of course the required RNP values must be met by the Actual Navigation Performance) 



If an approach requires tighter limits, such as RNP 0.3 or 0.15, then special authorization is required.
This is where SAAAR comes in (Special Aircraft and Aircrew Authorization Required), now renamed to RNP AR (RNP Authorization Required). RNP AR routes only make use of TF and RF leg types to fully fix the lateral path. RF legs are not supported by all navigation systems by default. 

To get authorization for RNP AR, the aircraft needs to comply with the applicable functional and performance requirements, the crew needs to be trained and lot's of documentation must be maintained. Basically proof must be provided that all of the three error terms are under strict control. 

In a sense it is somewhat similar to ILS CAT III approval. 



References: 

AC 90-101A describes a means of compliance for RNP AR 

RNAV / RNP presentation gives a good explanation of RNAV and RNP