A VPS is a virtual private server. It is a type of hosting plan in which you, the customer, has complete control over a virtual server. Often (and increasingly so) the VPS runs on top of cloud infrastructure (in other words — a large pool of networked computing resources, rather than a single piece of hardware).
One way to think about hosting plans is that a shared hosting plan is like renting a small apartment and a dedicated server is like a house. From this perspective, a VPS plan is like a condo.
With a VPS plan you get most of the benefits of a dedicated server, at a much lower cost. In the case of cloud-based VPS hosting, you also have the ability to scale up available resources quickly as the need increases.

Possibly.
There are two major reasons for needing a VPS hosting plan:

• High traffic — a few thousand visitors a day consistently, with the possibility of a lot more from time to time
• Highly customized configuration — often the case for a SaaS application, new development, or other projects that go beyond a strightforward blog

If either one of these describes your project, you probably need a VPS hosting plan. If both of them do, you definitely need a VPS plan.

That depends on what you mean by "better."
VPS is "better" from the standpoint that it is more customizable, has faster performance, and can handle more traffic than a shared hosting plan. Shared hosting is better from the standpoint that it costs less and is easier to set up.
A better way of asking this question is: Which type of hosting is more appropriate for me?
There are a lot of variables, but the simple answer is:

• Small, simple websites and blogs with a limited audience — shared hosting
• Large and/or complicated websites and web applications with a large user base — VPS hosting

Yes.
WordPress works well in most VPS hosting environments, and many are specifically configured to make it easy to install and setup a WordPress site.

Not only can you — you should!
WordPress Multisite works best on a VPS (or dedicated server), where the site administrator has full control over web server configuration and other environmental factors.
Also, the nature of a blog network, which contains many sites run by many different people, makes it somewhat difficult to predict things like traffic and resource usage. A scalable, or cloud-based, VPS plan will help you keep your network stable and running.

Yes.
VPS plans give you full access to the root user and sudo, ssh access, and the ability to configure your server environment as you see fit.

Almost certainly.
A VPS is a full-fledged server, to which you have complete access. Though it is a virtual server, not a piece of physical hardware, it is otherwise equivalent to having access to a your own, "real" hardware. You can install anything you want, configure the server anyway you want, even mess with the operating system.

A virtual server is a software emulation of an actual physical computer.

Generally speaking, VPS hosting is a little more expensive than shared hosting, but significantly less expensive than dedicated server hosting. You can compare pricing on our list of VPS hosting companies. Also, if you're looking to save money, be sure to get a web hosting coupon.

Possibly the most common virtual machine software for hosting providers is HyperVM.M