Lautering is the process of separating the sweet wort from the grain. The term “ lauter tun”is not something many home brewers will be familiar with because they don’t typically have a vessel between the mash tun and boil kettle. The wort is filtered from the grain through a filter media(sometimes it’s just the grain bed that filters) and straight to the kettle. Due to this a home brewers kettle will have more trub(thick gelatinous gunk, mostly proteins) than a commercial brewery. Due to the volumes, typically 5 or 10 gals vs 1000s, we’re less concerned.
Boiling the wort helps to concentrate the sugars in the liquid and to sanitize the wort of any bacteria or infection, but a long boil of 60 minutes or so is really used to extract buttering from the hops.
Pasteurization is mostly about stopping further fermentation and for sanitization and safety aka CYA. It’s seldom done on the home brew level.