Ok. Forget everything about the orbitals for a second.
You have nuclei. You have electrons around them.
First step: electrons are not balls. Imagine them as being a smeared charge in space. Which is precisely what they are: smeared charge in space, with fuzzy borders, like an unfocused blob. The shape of this blob depends from many factors, but the biggest one is the position of the nuclei.
Now suppose you want to describe the shape of this blob, like if you want to 3d print it. Well an easy way would be to divide space in little cubes and say "charge" if you have some smear or "no charge" if you don't have that smear. In practice, since it's fuzzy, you would say "3.0 charge" in some cubes, "1.3 charge" in other cubes, "0.1 charge" in other cubes and so on. This is a perfectly legitimate way of describing what is called "charge density" in space.
The problem with this approach is that it's rather inconvenient. It has poor accuracy and it scales badly. If you make the little cubes smaller, because you don't like living in a minecraft world, you need a lot more cubes to do that.
So now there's a smarter method to do so, and to explain it, I need you to understand Fourier decomposition. It's not as hard as you think it sounds.
The problem is the following. You have a complex sound wave, like the one produced by blowing a whistle or singing a song, and it turns out that you can create any complex shape of this wave by summing simpler waves together: a careful choice of sines and cosines of certain frequencies (pitches), and with certain intensities (volumes). Play them all together, you get the initial wave back.
This is exactly what you see in your spectrum display in your stereo
Where each column is a different frequency, and the height of the column is the intensity of that particular frequency. It changes all the time because you are playing complex music, but try to play a uniform sound (e.g. a violin playing a single note) and you will see it stays the same throughout.
Now back to the orbitals.
The orbitals are the "sines and cosines" of our problem of describing that blob. We have a complex entity (our smeared blob) and we want to describe it by summing together "something". It does not matter what "something" we use, but it turns out that 3 dimensional functions of that shape has a lot of nice properties which makes the problem much more compact.
Let's make you a simple example. Suppose you have a spherical blob of charge. That's probably described well by a single orbital of spherical shape (an s orbital), exactly like a sine wave from a tuning fork is described well by a single sine function.
Now add an electric field so that the electrons are pulled, and the smeared sphere is now more like an elongated egg. That one is not really described well by a sphere, is it? so you need to describe the lobe, meaning that you need an additional orbital (a p orbital) to add to the mix so that the result is egg-shaped, exactly like you need more than one sine wave to describe the sound of a violin.
That's it, really. orbitals are just convenient 3d equivalents of the sine and cosine. We could well use anything else (and in fact we do, in some cases) and it would work as well, but with some potential disadvantages.