Why is it just in 3D? 1D things making knots in 3D seems like it would have immediate analogs for m-dimensional things in n dimensions. Is that not the case? Is it not as rich an area of study or something?
It's been a while since I've studied this, but I believe it's the case that the only time you get nontrivial knots is when n = m + 2. The most well known case, of course, is when m = 1 and n = 3. But for every value of m >= 1, there are nontrivial knots in dimension m + 2. I believe it is indeed a rich area of research.
Essentially, knots (i.e. 1D things embedded into some other space without crossing itself) are trivial in dimensions lower than three because we don't have enough space to make it interesting. A 1D thing that doesn't cross itself in the plane must be a circle, warped in some way, but it can be unwarped without ripping the plane.
On the other hand, knots in dimension higher than three are trivial because we have too much space to work with. I don't know the details here, but this is what I've been told.
It's kinda miraculous that knots in three dimensions have such interesting and rich structure. Hopefully an expert will come by and give a more detailed explanation, but in the meantime, hope this helps :]
The more general field is called (Geometric) Topology, and includes all your m-dimensional objects in n-dimensional spaces.