Every level your character gains you can gain a level in a single class. Multi classing is when you split these class levels across multiple classes. For example when your character (let's say a gnome Wizard) reaches 5th level you decides to multi-class into a cleric. Your character is level 5, but more specifically a level 4 Wizard and a level 1 Cleric. (note: multi-classing is a little more involved than this, but this is the relevant part to your question)
Picking a favored class grants you an additional hit point or skill point for every level you take of that class. So if your gnome had wizard as its favored class, it would gain an additional hit or skill point for each of the first 4 levels, but not when it hit level 5 and took a level of Cleric.
Half-elves still level the same as every other class, but have multi-talented, which allows them to pick two favored classes. If our gnomish example character was a Half-elf, we would want to keep our favored class as wizard, but also make Cleric a favored class. Now when the character reached 5th level and took a level in Cleric, they would still gain an additional hit or skill point.