First of all, a little anatomy.

Sound is produced when the vocal folds vibrate while air is flowing past them. There are two vocal folds, about 1 to 1.5 cm in length each, made of muscle tissue, which vibrate back and forth very rapidly - several hundred times per second. As these two muscle groups become stronger, the voice becomes stronger. As they develop in flexibility, the finesse and control of the singer improves. These are small & delicate muscles, and it is very important not to strain or force them to operate. Of course when you are learning to use them in singing you will make mistakes, and overdo it from time to time, but over a period of time there should be increasing comfort as well as strength in using the vocal folds, otherwise you are not doing it right. (In a worst case scenario, like an untrained rock singer who yells, callouses called nodules start to grow on the vocal folds at the pressure points that form when they are made to vibrate roughly & harshly, and if not fixed by vocal training these nodules will eventually destroy the voice.)

The 2 vocal folds sit inside the larynx, which is also called the voice box or the adam's apple. Girls have the same structure as boys, but theirs is 40% smaller, and hence doesn't stick out as obviously as for males, so the adam's apple tag doesn't get used for females. (Incidentally, women are only 10% smaller than men on average, so it appears evolution made a big effort to make men's' voices sound different from women's').

The larynx is made of cartilage (similar to the nose) and has a degree of movement up and down the throat. All mammals tend to raise their larynxes on high pitches and lower them on low sounds. To feel this happening, gently place your middle finger across your voice box, so you can feel it but not interfere with its operation. Make some buzzy sounds like zzzzzzzzzzz to locate the centre of the larynx. Then yawn in a big way, and you should feel the larynx go down. Now cry out loud like on a roller coaster, and you should feel the larynx rise.
This shows that one can affect the level of the larynx through mental control. i.e. with practice you can think it into a high position or a low position. The high position makes getting top notes easier, but if the larynx rises too high it will choke off the sound. Lowering the larynx creates a warmer fuller tone, but can also take some of the brightness out, and will certainly prevent one accessing the very top range of the voice.

Enter Fido.

Suppose Fido was a pet puppy. Imagine this puppy is very excitable, and likes to jump up & lick people when they enter the room. We would want to train Fido to act appropriately. We wouldn't want to have to tie Fido down, or hold him down on the floor; we would simply wish him to respond to our commands not to jump too high or over-exuberantly; so we would train him over a period of time to behave. After all, he is a puppy, & it's natural for puppies to move about - we just wish to direct his energy, & not be overwhelmed by it.

Hence Fido the pet larynx.

The Happy Witch is an exercise which trains the larynx into a good position for high notes i.e. an open throat, with a high larynx, & a high tongue.

Start by laughing out loud hilariously, in your high voice, as though you were a witch who is delighted that one of her spells is working. Rub your hands with glee. Tee hee hee. This should sound very raucous & brilliant. Now laugh again & mentally give the action a number out of ten for the amount of effort needed to get that high energy sound happening. Try to get to a 6 or a 7, where 3 is feeble, & 10 is a deafening shriek. Next recreate the laughing sensation silently, but still keep the number for intensity in mind. Go back to an audible laugh, and then a silent hysterical laugh. At this point you should be able to feel that your throat is open, the larynx is high & free, & your tongue is not being swallowed into the throat. Now try some high scales with the larynx in this position. It will sound horribly shrill at first, but the notes should be strong & comfortable. Go up and down on ee & ay. With practice this exercise will give your high notes brilliance & ring, and eventually the harshness will ameliorate. (If you're lucky it won't be harsh at any stage. I'm not suggesting harshness is inevitable in a developing singer in the top range, merely observing that it is very common, especially for sopranos, & not a major cause of concern as the tone usually becomes rounder by itself with practice).

If you feel the tongue in the throat pushing downwards on the larynx, you can practice lifting the tongue out of the throat by pressing the sides of the tongue against your upper molars, & saying the alphabet out loud. Another good tongue exercise is lee ly low la lay sung up and down the scale on the notes c d e f g f e d c. lay is only sung on the last note. This working of the tongue loosens it up & brings it under the control of the singer. There is no one ideal position for the tongue, as it is used to articulate the various sounds in the language & has many different positions, but the higher the back of the tongue the better, as it makes no difference to the size of the space in the mouth, this space being so much larger than the tongue, but it can make a critical difference in the throat, where the tongue takes up a large proportion of available space.