Organized ice hockey provides an opportunity for players who want the entire experience of playing "real games." By real, I mean uniforms, teammates, a schedule, referees, timekeepers, and scoreboards.
When I started playing ice hockey, I first enjoyed pick-up games, and "rented stick-and-puck ice". For many players, a regular weekly scrimmage is enough. But in my area, Greater Boston, it was easy to take the next step and join a league because there was plenty of leagues to join (okay, there were two).
I played a season with Hockey North America (HNA). The league has a terrific beginner's program, and the first month was spent going through drills and having coaches evaluate the players for the beginner's league. I don't remember exactly how I was told, but after a few evenings of practices and scrimmages, I learned I would be joining the Boston Heat, a team comprised of beginner (D-level) players like myself.
When I look back on this experience, I think it was about as ideal an experience as I could have hoped for. It was very clear from that first meeting that we would have a good captain (Joe Jaena, RIP). As a team, we were very organized, and the first order of business was arranging to hire a coach and to rent ice for our own practices. The league didn't supply this organization, but it was pretty clear that having a coach and regular practices and some semblance of a plan kept us at the top of our standings.
The league provided the uniforms, a red jersey with a flaming puck. I still have mine (Umali, number 25). The league provided a schedule, and at the games, they provided a referee and a time keeper. At the many ice rinks I played at, the score was on a scoreboard. The games were often in the late evening hours, and as we progressed through the season, the game became that much more routine. Our coach eventually brought in an assistant, and we played (and lost) in the championship series.
New England has other leagues, and our team played at least two more seasons (once through the spring/summer, and another through the fall/winter). It should be noted that a league costs money, and by my recollection, HNA was more expensive that the regional league we joined later on. That said, if you're already outfitted to play, you already know that the game demands a certain financial investment.
As with finding stick-and-puck ice, use the people at your local rink (or rinks) to find out about your own area's opportunities for organized play. I can say without reservation that playing ice hockey "for real" was one of my favorite experiences as an adult.