Arguably any software which works with map data can be called GIS software, but the term is often associated with a breed of powerful "traditional" software packages, running as desktop applications, with rich mature feature sets which have been around long before OpenStreetMap existed (also see Wikipedia about "GIS applications"). Many are proprietary, closed source, and quite expensive. Often they prefer to operate with data in a proprietary data formats, but some formats have gained widespread support, for example ESRI Shapefiles and PostGIS databases.
GIS software and OpenStreetMap
Shapefiles and PostGIS databases can be created/populated with OpenStreetMap data. We should aim to document any special interoperability with OpenStreetMap for each GIS software package on the relevant wiki pages (linked below), but in general terms, OpenStreetMap data is Open! (vector and raster) As a result, people have worked out how to do a whole bunch of converting map data between formats including various methods for getting data into shapefile format. This often involves downloading data and then running separate conversion scripts.
In some cases, the more forward looking flexible GIS software makers/developer communities have created plugins or core compatibility with OpenStreetMap. Some GIS software will open a .osm file. Some use our raster tiles or WMS variants. Some have integrated with OpenStreetMap API. This is wonderful, and very much fits with our aim to get OpenStreetMap data used more widely.
Open Source GIS software
It's true! "Open Source" and "GIS software" can go together:
- QGIS (previously: Quantum GIS) () Free and Open Source. OSM plugins for importing. Discussion: wiki
- GRASS GIS ()
- gvSIG Desktop ()
- OpenJUMP ()
- MapWindow GIS 
- udig 
- SAGA GIS 
Some others listed at freegis.org
The other ones
- ArcGIS: Esri hosts the ArcGIS Editor for OSM , an open source extension to ArcGIS for downloading data from OSM into ArcGIS and supporting contributions back to OSM.
- MapInfo Some options for converting OSM to MapInfo Interchange Format (MIF). Also MID. Supports TileServers from 10.5 onward. 11.0 Premium includes access to Bing and OSM Tiles.
- FME - is an ETL for GIS data file storage (DB file) not a GIS software editor that supports reading OSM xml data (.osm) since build 5589 in 2009
- Smallworld (gepower.com) - Used a lot by utilities companies
- Maptitude Can import, edit, and export OSM data to many other formats. Also supports TileServers and OSM Tiles from version 2014 onward. 
- Manifold Gis System - Supports importing only from server osm data in raster  and vector  format since 2009. This data is not accessed directly as a raster or vector format, it is accessible using an object/gui API. Thus the vector data is stored in drawing object and raster to Image Object .Manifold could be used as an ETL like FME.
- Spatial Manager Desktop™ - Spatial Manager Desktop™ is a desktop application designed to manage spatial data in a simple, fast and inexpensive way. Includes its own OSM data provider
- Spatial Manager™ for AutoCAD - Spatial Manager™ for AutoCAD is a powerful AutoCAD plug-in designed for AutoCAD users who need to import, export and manage spatial data in a simple, fast and inexpensive way. Includes its own OSM data provider
- Spatial Manager™ for BricsCAD - Spatial Manager™ for BricsCAD is a powerful BricsCAD plug-in designed for BricsCAD users who need to import, export and manage spatial data in a simple, fast and inexpensive way. Includes its own OSM data provider
- w:List of geographic information systems software - more complete list
- w:Comparison of geographic information systems software - more precise comparison