nontechnical understanding of a mammalian expression system
The general principle of the expression system is simple: it is a biological machine that uses cell media as its input (i.e. a broth of simple nutrients, including the protein building blocks) to generate its output, a specific protein of interest, according to its &quot;settings&quot;, if you will, which is the DNA plasmid that describe which protein will be made. The cogs of the machine are carefully engineered mammalian cells that both create protein from nutrient, and multiply themselves to increase production capacity.
In a perfect expression system, there would be no need to know the underlying biology of the system: just provide the specified broth with a correctly formatted DNA plasmid, let it simmer for a specified duration at a given temperature, and voila!
The value proposition of a commercial system for its users is being as close as possible to this perfect system, i.e. not having to &quot;roll your own&quot; from first principle (after all, the general processes underlying protein expression are covered in an undergrad biology education) which would entail a great deal of effort and expenses. In some way, this is similar to the business model of cloud providers like AWS: abstract the low-level sysadmin-type stuff to provide a reliable platform for applications.
This is useful in both an R&amp;D context (you can iterate on the protein and specific process steps for your product, not on the expression system) and a production context (you have a reliable expression system, that conform to regulatory requirements, with vendor technical support).
However, even the vendors have not entirely abstracted the biology of protein expression, and the specific working of the system is part of the differentiating features of a specific commercial product versus other products in the same line, and competing solutions from other vendors. You have asked clarification on a few technical terms:

host cell lines: The carefully engineered mammalian cells, which have been tuned to product protein in great quantity, while maintaining quality (e.g. avoiding aggregates) and, typically, specific requirement on post-translational modifications necessary for the application the protein was designed for. It must be easily to culture in large batch. There cells are a blank slate in which the DNA plasmid will be inserted to start useful production. A vendor's expression system will feature a cell line that tries to fulfil these something conflicting requirements.
vectors: How to format the DNA plasmid (= protein recipe) and get it into the host cell line. For mamalian system this will typically be viruses that can infect the host cell line to deliver the plasmid. The reliability and ease of use of the vector system, as well as compliance with regulatory standards in a pharmaceutical environment is also a differencing feature of an expression system.
cell culture media: Specific cell line will have different &quot;nutritional requirement&quot;, as such vendors will have specific &quot;recipes&quot; for the nutrient broth in which to grow the cells of the system. This is more or less directly dependant on cell line, so less of a differentiating factor per se.
know-how: Biological processes tend to be messy, so some troubleshooting is expected even in commercial systems. Documentation as well as expert technical support can be an important selling point.

Ultimately you will need to understand to some degree the biology underlying the particular
expression system to be able to compare it horizontally to competing solutions from other vendors,
and vertically to other methods of producing proteins (e.g. prokaryotic expression system, cell-free systems), and, if you are in a customer-facing position, to be able to answer some domain-specific question on the spot. Here is an overview, here is another from a different vendor. I would suggest asking more specific question after perusing these.