What is embedding? I am new to this so do I need to learn topology in order to understand this? Cause I come across this which says that unlike the 2D sphere, 2d saddle surface cannot be embedded in 3D Euclidean space(source: http://www.astro.yale.edu/vdbosch/astro610_lecture2.pdf - page 16) which I have trouble understanding. Cause from my understanding, embedding is a description of a space as a co-dimension n surface in higher-dimensional space.

In that context of that lecture, they are not talking just about the general topological concept of embedding, but instead a more special  metric concept of embedding. A metric embedding is a topological embedding which preserves the metric tensor in an appropriate sense.
So yes, you do need to understand topological embeddings, but that's not enough, you also need to understand metric embeddings.
What the lecture is referring to (as noted in the answer of @Salvatore) is Hilbert's proof that there is no metric embedding from 2-dimensional hyperbolic space (the "saddle surface" in the lecture) to 3-dimensional euclidean space.
