Renown is a measure of a Garou's place in society. The more Renown he has, the more weight his words carry. Renown governs how quickly a Garou can gain in rank. Different auspices are also expected to gain renown in different areas.
Mechanically, renown comes in two varieties, temporary and permanent. Temporary renown can rise and fall over the course of a story based on how a Garou conducts herself. When she gets ten temporary points of Renown, she may petition an elder for a special rite (WTA that will recognize her achievements and convert that temporary renown into permanent Renown.
Permanent renown is more difficult to gain and lose. Usually only heinous crimes will reduce a Garou's permanent renown. Usually a punishment rite is required to reduce a Garou's permanent renown.
Once a Garou has gained sufficient permanent Renown, he may challenge a higher ranked Garou to try and gain a rank. Different auspices have different requirements for how much renown and in what categories they need to be before they can challenge for each rank.
Ragabash may pursue any type of Renown, but generally need more points overall than other auspices before they can advance. Theurge, Philodox, and Galliard favor Wisdom Renown. Philodox and Ahroun favor Honor. Galliard and Ahroun favor Glory.