Anjishnu Bandyopadhyay, PhD Student in Experimental Particle Physics from University of Bonn and
Stephen Selipsky, ex-particle theorist; Stanford Ph.D., research at CERN, BU, Yale · Author has 1k answers and 5.3m answer views
When you ask "what is something made of", you imply that this something is a composite system made of more "fundamental" or "elementary" components.
In most cases, this question would make sense. For example, a grain of sand is made of molecules. The molecules are made of atoms. The atoms are made of electrons, protons and neutrons. The protons and neutrons are made of quarks.
However, our current understanding is that elementary particles such as electrons and quarks are fundamental; they are not "made of" anything. Instead, they "make up" everything else!
Some physicists think that elementary particles such as electrons and quarks are made of even more fundamental things, called "strings". It's very important to stress that although this hypothesis is very intriguing and has led to the development of countless useful mathematical tools and models, it has not been proven experimentally.
To address your concerns in the question details, the strings in string theory are nothing at all like "normal" strings made of rope or some other material. They are called "strings" mostly for convenience. A more appropriate name would be perhaps "1-dimensional fundamental constituents", but "strings" is just more catchy :)
A very common question (that was also raised in a comment to this answer) is: "are string made of energy"? The answer is no. To explain, let's go back to elementary particles such as electrons, since they are more familiar to most people.
Are elementary particles made of energy? No, they have energy as one of their properties. You can measure a particle's energy. A particle also has other properties such as mass, momentum and spin. The particle is not "made of" mass, it's not "made of" momentum, it's not "made of" spin, and it's also not "made of" energy.
As an analogy from daily life, if you take a piece of metal for example, you can measure its properties, such as temperature. However, no one will ever claim that metal is "made of" temperature. This just doesn't make sense. Temperature is just a property of metal. Similarly, energy is a property of elementary particles. But it doesn't make sense to say that they are "made of" that energy.
If you are still not convinced, think about this: just like a piece of metal can be cooled to absolute zero (so it doesn't have any temperature), a particle can have no energy, at least in principle. So, what is a particle with no energy made of?
Addendum - on "classical" vs. "quantum" intuition and concepts:
I suspect that the problem most people have with coming to terms with the notion of a fundamental particle (or string, or whatever) is that the words "made of" have a certain common meaning from our daily life which just cannot be applied to physics at the quantum level.
The famous quote "no one understands quantum mechanics" means that we cannot understand quantum things using our "classical" intuition from our daily lives. But we understand it perfectly well if we use the new concepts and ideas introduced by quantum mechanics. And fundamental particles are one of these new concepts that we cannot understand with our old "classical" intuition alone.
The reason fundamental particles (or strings, if they actually exist) are called fundamental is that they are the point beyond which you cannot go any further to something "more fundamental". Let me illustrate this as follows:
What is a chair made of? Wood. What is wood made of? Carbon and oxygen molecules (mostly). What are molecules made of? Atoms. What are atoms made of? Protons, neutrons and electrons. What are protons made of? Quarks. What are quarks made of? Uh... Sorry, this question simply doesn't make sense because there is nothing more fundamental than quarks.
It's like asking "what is north of the north pole" or "what number is larger than infinity". These questions are impossible to answer because the questions themselves are meaningless.
(...Unless string theory is true, in which case quarks are made of strings, at least in some sense. But then the strings are the most fundamental thing, and asking what they are made of is again a meaningless question.)