Shibari means "to tie" in the Japanese language, which is a pretty good basic summary of what the practice is all about.
As the name suggests, it is a form of bondage play which revolves around knotted ropes, and has a strict set of rules regarding how to tie and arrange those ropes properly. The origins of the activity lie hundreds of years in the past.
Although experts tend to link it to the way that elite Samurai warriors were bound when captured, it eventually developed into a private erotic discipline (then known as "Kinbaku", which simply means "tight binding"). Shibari is a more modern term for the same set of erotic activities, which still involve restraint - though of a simulated kind.
Firstly, it helps to understand that we are talking about an elegant Japanese art form, not just a source of erotic pleasure (although it definitely is that, too). The aim of a seasoned Japanese bondage expert is to create forms of restraint which interlink in regular, mathematical patterns, and which also harmonize with the form of the human body. The way that the knots are positioned is also based around erogenous zones on the human body.
When applied skillfully, they exert pressure on key locations, producing intense forms of pleasure, and of pain. As with most bondage systems, there is a close interplay between pain and pleasure, with one informing and accentuating the other.
While standard bondage is usually designed to create a master-servant dynamic, in Shibari, things are slightly different. The person creating the rope network takes on a spiritual and artistic role, using their training to create harmonious arrangements which are tailored to every participant. And the person being tied is supposed to give themselves over to the experience, letting themselves be influenced by the ropes and the knots.
The result can be a keen sense of euphoria, or being "rope drunk" as some people call it. And it can also result in intense relaxation for some people, which isn't usually the case with mainstream bondage.
Although few people have the proper training to call themselves a Nawashi ("rope artist"), anyone can attempt Japanese bondage in their own home. To do so, you will need to purchase a special Japanese bondage kit, with ropes made from jute, hemp, or linen, which are between 4 and 6 mm in diameter, and around 8 meters long.
Then, you'll need to investigate the techniques employed. Not just any knots will do. To really create the right sensations (and employ Shibari safely), you will need to master some basic Hojojutsu knots. Tutorials can be found on the internet.
However, be warned: mastering them will take practice and time. But the rewards will be worth the effort.