So a league is just like a league in any other racing sim, they pick one or more cars, a set of tracks, and generally have 1 race per week at a specific timeslot.
Yep, a league, as in a hosted session setup, is always an 'unofficial' race. That means it doesn't count towards your iRating or Safety Rating, and hence your licence level.
iRacing's system allows, imo, a much better setup where they've made the 'pickup' races close, if not equal, to the quality (in terms of racing) of leagues whilst allowing the flexibility of racing whenever you're free, ish.
So when I say series I mean something like the "iRacing Mazda Cup", this is the rookie class road series. It uses the two Mazda MX-5's across a 12 week season (now at week 3 iirc), the week starts on a Tuesday and each week has a new track. You can do open practice with others at any time, qualifying is once per hour, time trials are every couple of minutes and races are also once per hour every day.
So instead of being fixed to Sunday at 14:00 like a league you can race then, or Saturday at 23:00, or Wednesday at 05:00. Whatever suits you. (assuming there are enough other racers, which does depend on series but I believe rookie races always have enough to run)