This is not a World War II reference.
Yes, I'm getting fancy and trying images. Phear my l33t Paintshop skillz, yo.
But, you know, the point
The theory behind the triangle is that the three components--scope, resources and schedule--make up any successful project. However, at any given time, you cannot change any of the lengths of the side of the triangle without impacting the other sides of the triangle in some way.
So, reducing the schedule will have an impact on resources and scope, whereas decreasing scope will affect resources and schedule...etc.
Examples: I have 2 resources on a 2 week project and a scope of work that can be completed in those two weeks. I receive a request to add additional features to the scope of the work. This effectively lengthens the scope side of the triangle. My options to manage this imbalance are to increase the schedule, remove scope from elsewhere in the project, or to add additional resources (or some combo of these three) to take the change to the project.
I liken it to explaining that you have a 5 pound sack. You can put anything you want in the sack that equals five pounds. Once you get to five pounds, one ounce, though, things won't stay in the sack--something has to go, a larger bag has to be gotten, or more than one person has to help carry (or some combo of these things). A lot of people defining scope for a project (or your team) can't actually see the sack, however. They don't know the size of the things they're asking you to put into the sack, and so they can't understand when the sack overflows, why it overflows, or how it affects their project.
Typically, people removed from your team don't know the implications of the changes they make to the work your team does. Your job, as a manager (and when you're managing a project instead of directly managing the people), is to communicate those implications in non-confrontational, clear-to-understand terms.
The triangle, and the 5 pound sack are methods for communicating the total amount of work that a team can take on. These metaphors are easier to understand that the complex calculations that you and your team use to determine how much work can be done in a specific time frame.
Now in order for the metaphors (and the team) to work, you need to know the size of the triangle and/or the bag, and the only way to know that is to work with the people who are represented by that image. Different methodologies use different ways to assess how many units of work can be completed in relation to time or task. I may go on more about these types of methods in another post, but in this one, I'll just suggest you review several different ways to do that type of estimation, then let the team pick the one with which they are most comfortable.
Then. Stick. With. It.
That was its own sentence because a lot of teams (a lot of companies) want to "try a lot of different things." The problem comes down to being able to consistently assess the amount of work over time with the available resources, and changing the measuring tool regularly makes that nearly impossible. Obviously, if something isn't working, change it or dump it, but everyone loves consistency: knowing what to expect can make work a much better place for you and your team to be. So, find something that works and stick to it.
Once you know how you are measuring things, asses the amount of work that you can do in a given week, two weeks, and a month. As noted, there are a lot of systems out there on how to do this, and they might even suggest you go further out to six months or a year. My generally experience has been that things get pretty shaky past two weeks of work estimation, and past a month your team is pretty much guessing in the wind. Your mileage may vary, but be aware.
Also, people who work with you and for you who have not done any estimating before in their lives will be BAD AT IT. Some people who have done it before are still bad at it; I had one Dev who responded to every request with "it'll take two weeks," and I would have to question him at length (and tell him jokes, etc.) to narrow down if "two weeks" was "two months" or "two weeks" was "two days."
Test out your methodology over time to confirm the accuracy of people's estimates and your own. Now you have the shape and size of the triangle and the size of the sack. Now you can tell people how much will fit, how much can be carried, how much can be moved around. You have these things in plain terms--you can write the thing out in numbers or draw pictures to communicate the limitations and boundaries of your team. In so doing, you protect yourself and your team from having their boundaries violated. This is huge for team morale, respect, and, you know, actually getting the work done. It also works extremely well to control scope creep and drastic reductions in schedule or resources because you can show people exactly what the effects of those changes have on the work at hand.
Ford once said that people could get "The Model T in any color, as long as its black." Being a manager (project or people) is much the same thing. The sack can only hold so much. Know what your team can do, offer that, and protect it against people who ask for their Model T with racing stripes.