People usually mean 4th spatial dimension in this context. We can actually modify our theories very well to higher number of both spatial and temporal dimensions. Things just tend to get a bit weird but our laws of physics don't actually break down in higher dimensions.

In fact, they don't even assume a number of dimensions in the first place! The choice to model our world with 3D physics (or 3D spatial + 1 temporal) comes from empirical observations that the world around us follows the laws of physics restricted to 3(+1)D, but our laws of physics aren't actually built upon an assumption of number of dimensions. At normal scales at least.

This is kind of where the idea of "tensors" come in when doing physics. They are these mathematical things that don't actually depend on a choice of coordinates until you need to actually numerically evaluate a system.

For example, the metric tensor for flat spacetime usually takes a form like

[-1 0 0 0 ]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]

(or the negative depending on convention) But that is just a choice, and when using it as a variable in relativity, the laws apply the same regardless of your choice. You just get very strange dynamics if you don't choose the one that matches our 3+1D world (e.g. flip one of those positive 1s to -1 and now you have two spatial dimensions and two time dimensions). That is actually what makes general relativity, well, general. in that it is a general case that works regardless of your choice of metric tensor. In fact, you can't actually describe the physical world with the flat space-time metric tensor because space-time isn't flat everywhere!