If we are talking baseball, the great American pastime, a franchise is a ball team, for example, the New York Yankees. The franchise, or team, is made up of players, managers and staff. A series is set of games that a ball team, (such as the Yankees), plays against another (specific) team, for example, the seven (7) games of the World Series are played by the two franchises (teams) that have won the most games in the regular season, (that is, the most games cumulatively from each sets of series between any two specific franchise teams)! Clear as mud?!

Can you be a bit more specific? Are you talking about TV shows or what? Anyway, without context, I would say a franchise is a lot broader. For example, there is a series of Star Wars movies (or even 3 series, if you want to argue that). But the Star Wars franchise encompasses a lot more: books, video games, comic books, even fan fiction, toys, lightsaber collectibles, etc. Even at smaller scale, It's Always Sunny in Philadelphia is a TV show that has been going on for ~15 years now. You could call it a franchise since it's big enough that they sell lots of shirts and other merch. Nowhere near the size of Star Wars of course, but bigger than a show in its first season.