Most modern jet engines used in large commercial aircraft use axial flow compressors. Typically the compression ratio over each set compressor disc is about 1.2 to 1. This is quite low but some engines might use up to about 30 stages of compression before the combustion chamber.
The compressor and the turbines that turn them are mounted on shafts, Rolls Royce powerplants are of a three shaft design and all other manufacturers use a two shaft design. Ths is because of patents. Each shaft counter rotates. The different shafts rotate at different speeds. The shaft at the core of the engine, ie either side of combustion chamber can rotate at tens of thousands rpm. Whereas the outer most shaft might only rotate at a couple of thousand.
The different rows of compressor blades are carefully aligned when then are designed so as to ensure that the airflow through the engine remains smooth, this is aided by devices called stators which are effectively static compressor blades.
If this sounds just a little too complicated I can sort some diagrams out for you.