There are only three dimensions by common usage of the term "dimension" as specifically "spatial dimensions".
The spatial dimensions are x, y, and z. You can arbitrarily relate these to east/west position, north/south position, and elevation if you so prefer.
If you ever hear of other dimensions beyond the spatial dimensions, they are either time (when you extend the definition of a dimension to be "one of the minimum instructions required to pin-point an event") or if beyond the three spatial dimensions and time, it is called a dimension of EXISTENCE.
There are several theories...string theory says there's 11 dimensions...others suggest 10...I've also seen 26 dimensions mentioned. There's also something called N theory that says there are "N dimensions"!!
Ultimately, it's unknown. Geometrically, there could be as many dimensions as there are numbers to denote them....the field would go on forever...both into the infinitely large and the infinitely small.
Although there is a formal definition for a dimension it all boils down to the minimum number of arguments N needed to locate a point p(x1, x2, ..., xN-1, xN) in time and space. N is the number of dimensions.
In our universe, we need 3 spatial dimensions and 1 temporal dimension to pin point a point in space and time. Therefore, we live in N = 4D space-time. As time is one dimension, that leaves N - 1 = 3D spatial dimensions.
When we remove one of the 3D, we are left with 2D. The picture on your computer screen is 2D because it lacks depth, it has only width and height. Width and height are the two spatial dimensions of your computer picture. Your cursor lives in that same 2D space, you can move it left-right, up-down, but not in fore and aft.
Some names we've given the three spatial dimensions are: x,y,z or i,j,k or rho, theta,r depending on the coordinate system we choose to use.
Pilots use position, time and altitude for the four dimensions to report where they are. There are three spatial dimension here because "position" invokes two dimensions like latitude and longitude. Add these two to altitude and there are the requisite three spatial dimensions. And, of course, time is time no matter what coordinate system we use.
And to answer "what do they look like," that's what a pilot's dimensions look like: position, time, and altitude.
You have heard of the so-called "higher dimensions" I suspect. Again they are simply arguments to pin point a point in time and space. But this time the space is 4 or more dimensions. We have no clue what they look like because we are 3D spatial beings and have no way to sense anything higher than 3D in space and the 1D in time.
The higher dimensions were contrived in some theories, like string/M theory, to make the answer come out right. Without the higher dimensions, they were getting silly answers like probabilities greater than 1.000. M theory invokes ten spatial dimensions. That's one more than string theory. Ed Witten needed to add that one more spatial dimension to merge the five string theories that existed at the time. Now, with M theory, we know the five were simply different points of view of the same things.