AHP has been used worldwide for over four decades now for decision making processes in various disciplines. As you know that there are a number of factors which influence landslides depending on the study area such as rainfall, slope, aspect, geology, soil, land use etc. Now, to make a landslide zonation map you will have to provide weights to different factors to see which factors have more or less influence on landslide causation. That is where you will need multi-criteria decision making which is based on AHP. In case you haven't seen, there is a nice example in wikipedia on AHP. If you just search for 'AHP' in Google you will get millions of results. I've added a few which are relevant. Alternatively, you can use ILWIS which is a free and open source GIS sw that has an in built multi-criteria evaluation tool based on AHP. Try it out.
There are little room to improve the detailed and supported answer provided by James but I think that one tiny valuable point is that AHP is both multicriteria decision-making tool and a ranking tool also, in this vein AHP could help you in ranking (perhaps in terms of risk of landslide) multiple zones within your geographical target region.
You can see my paper Arianpour and Jamali 2014 and 2015 in my RG page [Flood hazard zonation using spatial multi-criteria evaluation (SMCE) in GIS (Case study: Omidieh-Khuzestan)-and Locating flood spreading suitable sites for groundwater recharging through Multi Criteria modeling in GIS (Case study: Omidieh-Khuzestan)]. we gather opinion of experts in tables then we use Expert Choice software.