Vectors are tuples of numbers, an ordered list. The length of the list is the dimension of the vector: (2,3) is a two-dimensional vector where the first component is 2 and the second one is 3. (2,6,5,9) would be a four-dimensional vector and of course they can be arbitrarily large.

You can string together any number of values you want to keep track of in a vector. Length, width and height typically being used for three-dimensional space, but you can add time to that to create a four-dimensional vector. You can add more if you want, add air humidity for a 5th dimension, wind speed for a 6th one etc. it all depends on how much you want to record/keep track of.

edit: just realised I didnt really answer the question. To "solve something in n dimensions" means that there is a problem that can be formulated in such a way that it makes sense when talking about lists of length n. So for instance spheres (having a length, width and height) can be described using 3 dimensional points. A common way to describe spheres is "all points which are length r away from the 0 point". This naturally extends to other dimensions since we can measure length in any number of dimensions and thus talk about 4 dimensional, 32 dimensional or n dimensional spheres. Then we can try to solve the problem that was formulated in a certain dimension. The solution typically varies as more dimensions add more variety and using some quirks of certain numbers (prime/not prime, power of 2, divisible by 6 or whatever) its often possible to find a solution for certain dimensions, but not necessarily others. Having a problem solved for any dimension is typically the holy grail after which people will look into how make the problem more generic, or extend it in some other way.

So you can relate dimensions to the physical world, but the way we tend to think of them in math and physics is more abstract. Generally I think describing them as “degrees of freedom” is accurate; they are extra ways of viewing the functions or concepts you’re working with. Higher dimensions allow us to discern information about the lower dimensions we exist in, much the same way higher order derivatives show us behavior of the lower level function. Dunno if this is helpful I might edit this in the morning when I’m not sleepy to make it more useful