The vertical stack is one of the most fundamental offensive set-ups in ultimate. It consists of tthree handlers (1 and 7), and 4 vertically stacked, with respect to the field, cutters (3 through 6). The mark as shown is currently forcing a right-handed forehand.
For beginners, cutting may be daunting. It is especially hard to develop an intuition on when to cut and where to cut. The vert stack, at its most basic, is designed to resolve these issues. The vert stack separates the field into three major areas. With the current force set by the mark, the "open side" is the area to the right of the cutters (the area the mark is allowing the handler to throw; where the "open" throws are), the "break side" is the area to the left of the cutters (the area behind the mark; where the handler has to "break" the mark to throw to), and the deep space is the area above the cutters (requires a "deep" throw). These are the main areas cutters will cut to.
To address whose turn it is to cut, the cuts come one at a time from the back of the stack.
Once the cut is complete (generally one move), cycle back into the stack behind the handler in the stack (2). As 6 is cycling back into the stack, the rest of the cutters move towards the back of the stack. While 6 is cutting, 5 should already be setting up his or her cut, usually with some sort of fake.
Cuts don't always have to go to the open side. Break side cuts can often be useful, especially when you know the handler with the disc is capable of breaking their mark.
Deep cuts are also quite useful when you know your thrower has that sort of range.
When a cutter gets a disc, the handlers will cycle into the stack, depending on how they are positioned. It is usually judgment call by the handlers, but communication is key when doing this.
While moving up the field, the handler in the stack (2) directs the stack, mainly by setting a place for where the cutters will cycle back into, and for when the stack should move down the field. It may seem like this handler isn't doing much, but they are key when the person with the disc needs a dump. This process will be discussed in a separate play.
Once the cutter who got the disc up field passes it off, they will cycle back into the stack.
Once near the endzone (disc roughly within 15-20 yards of the line), someone will call out for an endzone setup.