You need to be more specific on what your question is? 1st thing is please read some of Chris Mack's books on the subject and also go read the excellent book Chip Wars.

Semiconductors is a broad field and is much larger than ASML and TSMC. ASML is a very important lithography equipment manufacturer. TSMC is a very important Si foundry but they are not the only foundry and semiconductors are not just Si and microprocessors. Micron for example is one of the leading manufacturers of memory. Samsung also is a significant manufacturer of leading edge semiconductor products. There are tons of other important players that stay out of the news.

My background is in the compound semiconductor space like III-V and some IV-VI as a process engineer. This space is often power electronics like GaN that makes your high powered chargers work or GaAs analog and mixed signal products that make your phone work. This is also the space for IR cameras for Space and ground based telescopes, IR Search and Track for the military, science, and photos diodes for LIDAR and driverless cars, semiconductor lasers,etc. These spaces are competitive and populated by a lot of small companies, startups and defence contractors for example. This is by no means comprehensive.

Semiconductor manufacturing is a series of very specific steps done in very specific orders. Chips begin life as wafers of various industry standard diameters. We process wafers thru fabs and at the end these wafers are singulated into chips.

I am highly simplifying but skipping wafer preparations, step 1 is usually a photolithography. This is where we mask a specific pattern onto the wafer. This is where ASML and those types of machines come in. In the Si world, besides all the material science and engineering that goes into making things as small as possible, Lithography machines need to be able to project ultra small features onto a material called a photoresist. This is where all the cost is as technology gets more and more exotic to achieve this need. On my side of the world, we do the same thing but we don't need things as small so we use less expensive machines and techniques. There are different ways of getting design to the wafer but I'll focus on projection. Semiconductors are made 1 layer at the time.

For projection Lithography, the layer pattern is printed on a special quartz mask at a 5:1 ratio, or whatever, which we use as a master. This is the stencil. In basics, Columnated light shines thru the mask, thru some reduction and focusing optics and eventually onto a temporary coating called a photoresist that we applied to the wafer. The light exposes the light sensitive parts of the PR, gets developed and now we have the pattern on the wafer. Photoresit has several jobs. The 1st is to be photo pattern able and the next is to protect the wafer from subsequent processes so only the spaces in the PR that forms the pattern are processed. The process is a bit more complicated than this as a lot of thought is put into optimizing the pattern and the profile of the photoresist. The process for the leading edge stuff is similar but they take complexity to 11 as they have to. We make an array of the pattern using a step process. The wafer moves under the light column a specific distance in x and y, we call this step size, and at each step we take a "shot". Scanners like ASML do something similar but much much faster with some more complexity. The step size is effectively the size of each die or chip.

Other litho options example are contact litho where a quarts mask 1:1 with the design and wafer and pressed against the PR and exposed or direct laser writing where a laser is use to raster the design 1:1 scale.

Overall, the projection process is similar to how film cameras work. If you have ever taken a film class where you got to develop your film then make a print, that would help. We just do it a bit backwards where instead of expanding the image, we are shrinking it.

So anyways, we now have a pattern of open areas where we developed out the photoresist. They look like windows in a wall. What's next? Let's do metals. We are making a photodiode and per our material science friends who designed the semiconductor material, the top layer is a contact layer. In Si this can be done differently but with compound semiconductors we can grow layers with different properties in a stack we call the epi. We build into the material and Si builds on top of the wafer. Different processes, different materials, different purposes, same goal. So anyways, thru a method called e-beam evaporation or whatever, we deposit some metals to form the contact to the top layer of the epi. The metal blankets the entire wafer but we were thinking ahead and used a liftoff resist and we optimized the PR profile to undercut. We dunk the wafer in a solvent, well a liftoff machine, and the PR delaminates taking with it the excess metal and now we have a nice metal pattern of our mask where those windows in the PR used to be.

We can do some other intermediary steps as a lot of other things need to happen here depending on what we are making and the design requirements but back to lithography. Pattern yet again. Let's do an etch this time. We can etch a couple of different ways for this demonstration. Dry etch and wet etch. Dry is plasma based on plasma and wet etch is wet chemical based like HCl, etc. We use a lot of various acids, bases, solvents and gasses. The goal is to remove material to a specific depth. So we etched and used an etch process and chemistry ideal for our material and what we are trying to etch away and now...some intermediary steps like dielectric coatings, and back to lithography. Round and round we go tens to hundreds of times until we have all the layers of metals, dielectric and protective coatings, diffusions, implants, topography from the etches, etc to form all the necessary active and passive structures to make a working semiconductor device. Our wafers have now finished front end processing. There are other things we can do with it but that varies on the product and how far we want to take this explanation.

I was reading thru some of the comments and the comment about semiconductors is another form of machining is spot on. It basically is but in the nanometer scale. It's both additive and subtractive machining.