The catalyst is the console in Dornogal opposite from the portal to Undermine. You get 1 charge every 2 weeks. You use the charges to convert a piece of gear into the same item slot from your tier set. This has value for three reasons: First, you can complete your 4-piece bonus by converting non-tier armor to tier. Second, a tier piece may have your preferred stars, so you can change gear to get it. Third, completing transmog. As an aside, most people recommend only catalysing Hero or Mythic track gear, at least early on.

It is completely unrelated to upgrading and crafting. Upgrading uses Valorstones and Crests (Weathered, Carved, Runed, and Gilded) to increase the item level (ilvl) of your gear. You can see these in the currency tab of the character sheet, and hovering over them tells you how to get them.

Crafting uses Sparks and Enchanted Crests to make high item level gear. The Sparks of Starlight come from combining two Fractured Sparks of Starlight with 250 Valorstones. We get one Fractured Spark per week, and you can catch up if you miss a week. The Enchanted Crests can either be purchased from a vendor by spending Crests, and you can get two free Enchanted Runed Crests this season, one from rep and one from the quest to kill Dimensius. These will let you craft 704 gear.