You are not alone in your confusion about f-stops and shutter speeds. The answers you have received prove that.
This explanation is lengthy and technical. There is no other way to explain shutter speeds and f-stops that I know of. You may want to print it out and review it and make notes. All examples used are hypothetical and unlikely to be duplicated in the real world. However, the relationships shown are valid.
A correct exposure is the result of the interaction of four factors: Light, ISO, F-stop and Shutter Speed. Having four variables can result in confusion if we start changing them willy-nilly. So lets eliminate two of them: ISO & Light. We are going to set an ISO and leave it alone. Our Light will be daylight (which can be manipulated but that's best ignored for now). Now we have only F-stop & Shutter Speed to deal with.
Here is a simple definition of ISO, F-stop & Shutter Speed.
ISO is a measurement of the sensitivity of a receptor (film or electronic sensor) to light. An ISO of 50 would indicate very low sensitivity; an ISO of 800 would indicate very high sensitivity.
F-stop is a term used to describe the size of the opening formed by the diaphram inside a lens. A small f-stop number, such as f1.4, would be a very large opening. It will admit all the available light there is. A large f-stop number, such as f16, would be a tiny opening and admit very little light. The f-stop we choose also determines shutter speed. Also, the f-stop we use determines what is called Depth of Field (DOF), loosely defined as that part of our scene, in front of and behind our subject, that appears in focus. We will have more DOF at f11 than at f1.4 but the actual "why" of that isn't important now.
The shutter speed determines how long our light-sensitive receptor is exposed to light to make a correct exposure. It also controls "stopping action" as when kids are running and playing. The higher the shutter speed the more action-stopping it is but the more light it requires so we have to use a larger f-stop (f1.4, f2, f2.8, etc.). Back when I learned all this cameras were mechanical and shutter speeds were expressed as fractions of a second - 1, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000, 1/2000. Todays electronically controlled shutters are called "stepless" which simply means they might be 1/50 or 1/70 or 1/987. For our discussion I'll use the old shutter speeds to illustrate the f-stop/shutter speed relationship.
Take a five minute break here and clear your mind.
Now lets see if we can put all this together. We're outside, its a sunny day, and the kids are playing. We've set our ISO to 200 and our camera in Aperture Preferred Mode (we choose an f-stop, the camera chooses a shutter speed. I shoot almost exclusively in this mode).
Our light and ISO are now "set variables" - we're going to leave them alone. The following is purely hypothrtical and for illustrative purposes only.
Timmy is running and jumping and we want to capture this as a "frozen moment" - in other words with no blur. We set our f-stop to f4 and the camera sets a shutter speed of 1/500 second. Timmy is "frozen" in mid jump. Timmy, however, is really fast and we notice his feet are blurred. We decide we don't like that so we change our f-stop to f2.8 and now our camera chooses a shutter speed of 1/1000 second.
You're probably wondering why or how this occurred. Remember that the f-stop controls the amount of light entering the camera. By "opening up" one f-stop (from f4 to f2.8) we admitted twice as much light. In order to properly expose our light-sensitive receptor the shutter speed has to increase - 1/1000 being twice as fast as 1/500.
Now we decide that we'd like to show some blur as Timmy is running. Our "static" photo above just doesn't show action and we know that inducing some degree of blur will show a feeling of motion. So we set our f-stop at f5.6. Our camera will now use a shutter speed of 1/250. Why? Because we have now allowed only 1/2 as much light as we did at f4 - and our shutter speed has to be slower to insure proper exposure. Since we like to experiment, we decide that even more blur might better show our little speedster Timmy and how fast he is. So we try f8. Our shutter speed is now 1/125 of a second. Again, f8 admits 1/2 as much light as f5.6.
Take five.
Lets look at a hypothetical chart which uses the above examples. Again, ISO 200, sunny day.
f1.4 @ 1/4000 sec.
f2 @ 1/12000 sec.
f2.8 @ 1/1000 sec.
f4 @ 1500 sec.
f5.6 @ 1/250 sec.
f8 @ 1/125 sec.
f11 @ 1/60 sec.
f16 @ 1/30 sec.
Every time we "stop down" - from f1.4 to f2 to f2.8, etc. - we admit 1/2 as much light and our shutter speed decreases in order to properly expose our light-sensitive receptor, be it film or sensor.
All eight of the above exposures are IDENTICAL. Yes, i said IDENTICAL. The only differences we'll notice are:
A) better image quality due to the fact that lenses are optimized for maximum image quality somewhere between f5.6 and f11 and
B) an increase in DOF from f4 all the way to f16, with the most DOF at f16.
In the situation you described, at sunset when the light can be intense in some areas, I assume you have shadow areas that the kids are running in and out of. Under those conditions, you might want to leave the f-stop at f4 and pay close atention to the shutter speed your camera selects. As our subject moves from light to shade with the f-stop set at f4 our shutter speed may drop to 1/125 sec. due to there being less light available in the shade.
Take five.
Under the above conditions - sun to shade - we might decide that our shutter speed is more important than our f-stop. Here we'll change from Aperture Preferred to Shutter Preferred (we set a shutter speed and the camera selects an f-stop for correct exposure). Now we can set our shutter speed to 1/250 sec. and let the camera change the f-stop as our lighting conditions change. However, we need to pay attention to our viewfinder to make sure we don't exceed the camera's ability to choose an appropriate f-stop. If the camera "runs out" of f-stops, i.e., at your pre-selected shutter speed of 1/250 sec. correct exposure is impossible even at f1.4, there should be a flashing light (usually at the bottom right of your viewfinder) to indicate under exposure. If this happens we must decrease our shutter speed to 1/125 sec. There should also be one at the top of your viewfinder to indicate over exposure. Your Owner's Manual should fully explain this feature if your camera has it. The same warnings will appear whether we're in Aperture Preferred or Shutter Preferred.
If you have questions just email me.