We live in what most commonly call a 4-dimensional world. We refer to the spatial dimensions in our world as length, width and height. The fourth dimension is really not a dimension but a critical component of the other three, it is time. Time is necessary to describe objects in those 3-dimensional planes and reference locations. The three spatial dimensions are at 90 degrees relative to one another.
Additional dimensions could provide a whole new world where things happen that we could not understand and could very well contain the spirit world, angels or demons which could be lurking all around us, unseen but very real. Angles or demons could be tracking our every move and actually be the proverbial fly on the wall. They could be carrying on conversations but we do not possess the senses necessary to detect their conversations.
I am reminded of a particular conversation in Bible text, where Abraham is discussing the dire situation Lazarus was in, Luke 16:26: "And besides all this, between us and you there is a great chasm fixed, so that those who wish to come over from here to you will not be able, and that none may cross over from there to us". Interesting how there is an apparent barrier or impossibility to connect to the other realm.
These additional dimensions, while impossible for us to see or perceive could still exist and would most likely help explain some of the more bizarre effects we detect in the sub atomic world. It could help explain gravity or even something Einstein called "Spooky Action at a Distance", otherwise known as quantum entanglement, which can occur when two particles are separated yet with an ability to communicate the other particle's state, instantaneously.
Physicists have proposed a theory called String Theory which seems to explain our world pretty well. It requires up to 11 dimensions with the smallest components called strings which would vibrate and the manner in which they vibrate, determines the type of particle they make.
A string is so small, they would be a millionth of a billionth of a billionth of a billionth of a centimeter. If you inflated the size of an atom to the size of our universe, a string would be the size of an average size tree. Consider there are approximately 20 trillion (20.000,000,000,000) atoms in an average grain of sand.
Viewing additional dimensions:
To understand how additional dimensions could exist yet not be perceived by us, consider the graphic below of a 2-dimensional red ball which has height and width. A two-dimensional person would be able to see the ball moving back and forth, however, it is a completely different to a one-dimensional person.
A one-dimensional person would not be able to see or perceive width or movement back and forth. To a one-dimensional person, they would only be able to see the height of a line and the height increasing and decreasing as the circular object passed before them (middle of image) and their field of vision (dotted line). To the one-dimensional person, a circle is impossible to process and only exists in the minds of the one-dimensional heretic.
Physicists at the Large Hadron Collider increased input energies in 2016 anticipating evidence of additional dimensions when proton beams collide and particles possibly disappear before the detectors. It may be sometime before the data will reveal evidence of an occurance.
Some physicists and scientists have proposed that there could be as many as 11 dimensions, however just trying to add a fifth spatial dimension is difficult enough to explain let alone visualize. To provide a small demonstration of how complex multiple dimensions can become, the following video illustrates how a possible 4th dimension could be viewed and visualized by a 3-dimensional person. Note that when we throw time into the mix we would have 5th-dimensional space-time and also, a 4th-dimension could possibly extend an infinitesimally small distance, totally beyond our perception.