This question feels fairly open, especially as we already talked the structure of a Web shop through quite comprehensively on multiple days.
Structure
Shops are not the only type of ecommerce websites out there. Depending on the type of the service offered, different structural elements may be the central point.
Example: online banking.
Accessing and managing existing assets (the bank account, saving accounts, loans taken, investments etc.) is much more important than new things for sale. Obviously the bank will want to upsell but when the customer cannot reliably use the main functionality, itās a massive deal breaker in more ways than one.
Example: custom-designed goods such as wedding invitations.
Communication channels and data exchange or collaboration tools between the customer and the designer are much more important than in a regular shop.
Example: push content such as an online course delivered via e-mail (I swear Iāve still seen some as recently as last year!).
There is often little in terms of presentation or itās just a typical blog. The push mechanism, scheduling and possibly communication channels between the instructor and the participants are key.
Example: tax or social security filing.
Not much in terms of presentation here but if supported, integration with other state service and an API for interfacing with popular bookkeeping software would be central. Validation is crucial, wizards important if offered. Also a good overview of filings made with a legally valid receipt feature and a document export feature.
Data
Customer accounts: addresses, means of payment, preferences.
Products: unstructured descriptions, images and other multimedia, comparable/searchable/filterable specifications.
Reviews: both unstructured comments and quantifiable (stars, but also āThe sizes of these dresses are small / large / as expectedā), possibly images.
Commercial analytics data, e.g. typical order value, products bought together, most and least popular categories and products, A/B test results.
Performance analytics data.
Usability analytics data.
A minirant on content here: as testers we may or may not be involved in the phase where actual content is displayed rather than some lorem ipsum tailored to various test conditions.
Pity because a lot of the descriptions just beg for someone with a decent command of the language to look over them, and itās not like there are no proofreaders and editors out there whose job is exactly like ours but for language 
Non-technical factors
But not everything can be ensured in the technical way alone. Often as testers we get to work with placeholders and obviously the human elements of the process / user experience will not be under test from us. But it can very well make or break the success of an ecommerce website.
Finally and as a side note, not a testing suggestion: if the goods offered are unique or of really special quality (think local, organic or with a special emotional meaning), customers will suffer through a lot and jump through a lot of hoops to get the shopping done 
