Modern professional maps, such as the one you linked, are made using GIS (short for Geographic Information Systems). To be blunt, this is not the kind of thing you'll likely be able to do as a hobby, although you're welcome to try. This is the kind of thing people get graduate degrees in. It's not extremely complex, but there is a really steep learning curve and you'll probably find it counter-intuitive at first.

GIS allows you to input, store, manipulate, and display different kinds of information that is spatially referenced. If you've used any image manipulation software (such as Photoshop), you'll notice some similarities. The key difference is that the layers in a map document are all spatially referenced to a coordinate system and displayed in a particular projection. Often the cells or vector objects displayed in GIS contain many different kinds of information stored in attribute tables (think Excel tables, but again spatially referenced.) For example a layer of points denoting historic/archaeological sites will likely have an associated attribute table listing the name, size (area), time period, and associated culture for each site. You could then use SQL commands to, for example, select only those sites that are over 50 ha in size and occupied between 200 and 800 AD and display them in a map.

Like I said, it's complicated. The go-to software is ArcMap (made by ESRI), but it's prohibitively expensive if you're not using it for a professional application. The open source alternative is QGIS which you can get for free. Both require pretty good computers though, and be prepared to experience lots of frustration figuring out why something isn't working only to realize you forgot to check a box when importing a file.