When working Scrum we are familiar with the words "Product Owner" and "Product Backlog" but what is "Product"?
The Product is the thing that you are building, or the service that you offer, or the need that you want to satisfy, or the vision that you have. The Product is first and foremost an idea "to do something". Only then you can you make it happen. And only then you can decide how you will make it happen.
Like putting together a Scrum team, for example. You might use Waterfall, Kanban, or some other way, it doesn't matter, but for the sake of this post, you decided to go with Scrum.
An now that you have chosen a way to do it, some people need to be responsible for getting stuff done. The developers will write the code, testers will test, designers will design, the Scrum Master will make sure you have some good processes, tools and techniques in place to support the work, and of course someone needs to take the thing you are building, offering, or the vision and split it up in little tangible pieces the people can work on and have a visible outcome that shows you are getting closer, step by step, to the thing you want done. That person is the "Product Owner" and the list of items to do is the "Product Backlog".
But I still haven't really defined what the Product is, right? And that's because there is no general definition, unless you take one from the dictionary in which case it means something physical that can be manufactured to sell for a profit, or the result of some activity that fulfills the purpose of embarking in the activity in the first place.
A product is different for each company, that's why it's hard to define. It's what you do, it's what you sell, it's what you offer.
Mike Cohn has a blog post that I think will do a better job at explaining it than I can, so check it out: What is a Product?
EDIT: as of the 2020 version, the Scrum Guide provides a definition for product:
A product is a vehicle to deliver value. It has a clear boundary, known stakeholders, well-defined users or customers. A product could be a service, a physical product, or something more abstract.
I like to think of a product as having a distinct set of end users who derive value from its use.
end users -> People that derive final value from the product, not people that will use the thing to make another thing (i.e. it is not a component)
distinct set -> Identifyable group of people get value from it, not 'the company' or some other ambiguous entity
derive value -> If the product did not exist then the value would not exist, so a process or a methodology is not a product
A product represents the promise of future delivery of (self)-service, via use of that product.
When we obtain a product, we trust that that product will deliver on its promise of service-delivery at the required time.
Bogdan gives a fair answer (saying basically the definition of a product depends on the company - i reckon it's meant, more specifically, the terminology agreed between the decision makers) linking to an interesting article with the same question but different take. Still, the answer uses the definitions of the entities product and service as synonyms, which even though it can be true i believe it requires a little more attention (this could possibly be another answer to another question "What is the difference of product and service?").
Here’s a product:
…used in a service (the services of a pizzeria) to create another product (a meal on a plate) for another (self)-service (eating the meal)
— a service is a means to deliver value
— a product is an asset held between services, that is output by one service for future use in another service