A template website is known as a website that uses templates that are repeatedly sold by a third party. Generally and hopefully (so you're not being ripped off) these are sold for a low cost, but isn't always the case (see related article).
The purpose of a template website in the eyes of the seller, is to create a website that superficially looks good and offers baseline functionality that many businesses are after that can be repeatedly sold as is - it's a one size fits all, off the shelf model.
In short, scope. A template website in a best-case scenario will provide multiple page template designs and functionality for a low cost.
If you're a new business and can't afford a professional solution, you can at least get your business on the web and start advertising/selling, assuming you don't run into any issues, which are more common than not from most providers.
Template websites are typically developed to a budget by the creator. They focus on maximising the functional scope of the offering, however they have limited time in which they spend to create the broad functionality so that they can make a profit based on the multiples that the template is expected to sell. This typically results in buggy applications that do not work as expected due to time versus scope being spread too thinly.
Due to template websites being sold in multiples (anywhere from dozens, hundreds or thousands of times), it's not uncommon to ultimately stumble across another website that has the exact same look and feel. The layout will be the same, the overall colours and imagery may still be the same or have minor modification, and generally just the logo is changed. It's easy to spot and when seen by a consumer, will devalue the brand of the business.
A template website will almost always cost you more. It's uncommon for a template website to be correctly set-up for a business' requirements right out of the box. Typically, a business will want the look and feel of the site customised, to add functionality that isn't included or sometimes simply get the thing to work. What's worse is that when those modifications get underway, they'll commonly take more time to fix or modify due to bloated and poorly structured code which as a user you don’t see.
The cost of these changes can add up fast and it can be better just to get an experienced web professional to create a custom production from scratch in the first place. The web pro will cost more than a template website upfront, however their solution will be superior and will save you time, money and stress from complications that template website projects commonly deliver in the long run (see related article).
Like always, there can be exceptions to the rule as in all things. However the observations in this article reflect many client first hand experiences who, after being burned by templates have jumped ship to working with us due to the reasons above.