These are terms typically used in the gaming industry. We use them to discuss network effects businesses because they highlight an important distinction.
A single-player product helps the user alone, and can be used without other users there. The user gets value from a product/service in isolation. For instance, buying something off Amazon. Or filing your taxes on TurboTax. Or implementing Stripe on my ecommerce site. It works for me right away, but I get no value from the fact others are also implementing Stripe on their sites.
Users pay for a single-player product based on the value it provides them directly, which tends to be linear. All the burden is on you to make your product increasingly more valuable to your customers. This is true of most SaaS companies like WorkDay or Oracle.
"Multiplayer" products, by contrast, let their users feel the presence and impact of the other users of the product, but have a hard time being useful without those other users. Multiplayer products let users add value to each other through their actions, even if it's just viewing something, like on YouTube. Vimeo, without comments and without video view count, is a single player experience. YouTube, with view count and comments, is a multiplayer experience.
It is possible for products to have single-player value and multiplayer value. For instance, HoneyBook has a tool that lets event planners make more beautiful proposals for their clients. Over time, as more photographers, caterers, and florists join the HoneyBook network, the event planner benefits from the professional network and N-sided payments network that begin to accrue.
It is also very possible for companies to add multiplayer mode to a single-player product, or even to switch their products from single-player to multiplayer. We help companies do that frequently. For instance, Amazon added multiplayer experiences with reviews, allowing past purchasers to benefit future purchasers. They also added a marketplace where hundreds of thousands of sellers joined and began adding value to buyers. For the first time in their history, Amazon's Marketplace accounts for over 50% of the total units sold according to the 2017 annual report. This is a huge shift.
We don't suggest building single-player-only businesses. They tend to grow linearly and get dragged down by competition.