Bare-metal servers have a single 'tenant'. They are not shared between customers. Each server may run any amount of work for the customer, or may have multiple simultaneous users, but they are dedicated entirely to the customer who is renting them. Unlike many servers in a data centre, they are not being shared between multiple customers.
Bare-metal servers are 'physical' servers. Each logical server offered for rental is a distinct physical piece of hardware that is a functional server on its own. They are not virtual servers running in multiple on shared hardware.
Development of virtualisation
At one time, all servers were bare-metal servers. Servers were kept on-premises and often belonged to the organisation using and operating them. Operating systems developed very early on (early 1960s) to allow time-sharing. Single large computers, mainframes or minis, were commonly housed in centralised locations and their services shared through a bureau. The shift to cheap commodity PCs in the 1980s changed this as the market expanded, and most organizations, even the smallest, began to purchase or lease their own computers. Popular growth of the internet, and particularly the web, in the 1990s encouraged the practice of hosting in data centres, where many customers shared the facilities of single servers. Small web servers at this time often cost more for their connectivity than their hardware cost, encouraging this centralisation. HTTP 1.1's ability for virtual hosting also made it easy to co-host many web sites on the same server.
From around 2000, or 2005 in commercially practical terms, interest grew in the use of virtual servers and then cloud hosting, where Infrastructure as a Service made the computing service the fungible commodity, rather than the server hardware. Hypervisors were developed which could offer many virtual machines hosted on larger physical servers. The load pattern of multiple users has long been recognised as being smoother overall than individual users, so these virtual machines could make more efficient use of the physical hardware and its costs, whilst also appearing to have higher individual performance than a simple cost-share would suggest.
Hypervisors provide secure isolation between tenants but there can still be a 'noisy neighbour effect'. If a physical server is multi-tenanted, peaks of load from one tenant may consume enough machine resources to temporarily affect other tenants. As the tenants are otherwise isolated, it is also hard to manage or load balance this. Bare-metal servers, and single tenancy, can avoid this. As, once again, server costs are dropping as a proportion of total cost of ownership against their administration overhead, the classic solution of 'throwing hardware at the problem' becomes viable again.
Bare-metal cloud hosting
Bare-metal cloud servers do not run a hypervisor, are not virtualised -- but can still be delivered via a cloud-like service model.— 
Infrastructure as a Service, particularly through Infrastructure as Code, offers many advantages to make hosting conveniently manageable. Combining the features of both cloud hosting, and bare-metal servers, offers most of these, whilst still conveying the performance advantages.
- Reynaldo Mincov (July 25, 2014). "Bare metal vs. virtual servers: Which choice is right for you?". Thoughts on Cloud.
- "What is a Bare Metal Server?". Rackspace.
- Eric Sarault (February 26, 2015). "Bare metal vs. hypervisor: The evolution of dedicated servers".
- Adrian Bridgwater (September 6, 2013). "What is bare-metal cloud?". Computer Weekly Application Developer Network.