That is simply caramelized sugar that coats the dish before the custard is poured in for baking. As the flan is chilled, the moisture from the custard liquefies the caramelized sugar and results in the sauce.
If you're wanting to make it for other desserts and aren't making flan, then simply caramelize sugar and then add water to it. Be careful as the steam can easily burn you if you're standing over it. The sugar will seize up and then as it continues to cook the crytallized caramelized sugar will liquefy. You will then have caramelized sugar syrup.
I am not sure what your specific problem is, so here is some general advice on making candy and caramel.
As for control, the normal method is to control it with a thermometer. (They are even called "candy thermometers"). The thermometer should have a very fast reaction time, and be capable of measuring in fractions of grades (one digit after the decimal point is enough even for Celsius measurements).
You should also have a source of heat which is as responsive as possible. Induction and gas work best. If using resistive electricity, be prepared to yank the pan away from the heat a degree or two before the temperature limit is reached (you must calculate some carryover: The heat from the pan will continue to heat up the syrup even after the source of heat has been removed. Expect more carryover from copper core than pure steel, and more from steel than from aluminum).
The caramelization of pure sucrose begins at around 160°C, but if you mixed in other types of sugar, it is different. For any type of suger, with rising time and rising temperature, you get a darker, less sweet caramel. It is up to you which kind you want, but the very light kinds don't have the complex aroma of the middle hued ones, and the very dark ones are somewhat harsh.
There are charts for the stages of sugar syrup (threads, soft ball, etc) and the corresponding temperatures. They determine not only the degree of caramelization, but also the hardness of the finished candy. It is easy to find an English language one with Google, try this one. A bit of search should turn up some which contain photographies of the candy instead of penciled illustrations. Use such a chart when making candy, you need it.
Another variable in candy comes with your ingredients. Acids will soften caramel. Enough of them will keep it liquid. Milk and cream will soften it too. Pay attention to when and how you are adding them. If you let dairy products sit in the pan for too long, they'll develop their own scalded milk flavor. Also, don't add cold liquids to hot caramel, it will seize up.
That's about it, if you can point out specific problems you are having, it will be easier to help you.
Best Answer
A medium saucepan would probably be in the 8-10 inch range.
Your sugar isn't colouring because the heat isn't high enough or it's high enough but not long enough. Just keep cooking, the change will happen. You can try less water but if you're not familiar with working with sugar, better to err on the side of caution; I do work with caramels a lot, and when I know I'm liable to get distracted by other things I have on the go I purposefully use more water than needed to give myself breathing room.
Are you using the correct size of dish as specified by the recipe? Some of the caramel will dissolve into the filling as you pour it in and as it cooks; I wouldn't worry about the thickness at this stage.
Beyond that, I would personally stay far, far away from anything done by RR. I don't criticise cooks lightly, and I generally try to cook some of their recipes before I'll pass judgement. The recipes of hers that I have tried, both from 30 minute meals and the website, are uniformly awful. Poorly written, ingredient quantities are off, and generally they just taste nasty.