Though it is different at each park, and changes from time to time, at park opening there is literally a rope. The guests are allowed into the park some time before the official opening time, but only up to the rope. There is usually a ceremony of some kind - quite entertaining in many cases - and then the rope is "dropped" (actually pulled to the side so nobody trips - but it's called dropping by analogy).
sometimes a "land" opens up a bit later than the general park. There is rope drop for this as well. Sometimes with a flourish, but not as big a ceremony as the park opening.
Sometimes there is a line of case members (CMs) holding the rope, and they walk along a path back to a popular area or attraction. This is often done if there have been incidents of people running dangerously back to an area or attraction in an attempt to be the first of among the first to arrive. This often creates a sort of stampede situation, or at lease a dangerous race of sorts, and the walk with the rope is a way of avoiding the danger and therefore avoiding injury.
The experience at rope drop varies by park. However, I would definitely plan on making the opening ceremony for the Magic Kingdom. It is charming and very entertaining. I can't remember how long in advance of opening time that it starts or what time to recommend arriving. The days where MK has morning EMH, the opening differs, or may be skipped, so plan on going to MK opening on a day that doesn't have morning EMH. I seem to remember planning on arriving 45 minutes or so early, with the opening ceremony beginning about 20 minutes before rope drop. I'm not sure about the times - maybe another poster can help, or you can ask your travel agent or a disney reservation agent or someone to be sure.
It's really a lot of fun - a must-do for us every trip. There is a little musical stage show at the train station rail facing the entrance, then for a finale, the train arrives with all the top disney characters on it waving to the crowd, then a patron, usually a young boy or girl from a family that was selected earlier gets to officially open the park by throwing pixie dust over the railing! Guaranteed to bring tears to your eyes every time. I'd recommend it highly!