I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would like to ask for a "take home universal informal definition".
Of special interest, as a physicist, is the instanton-Floer homology. But what mathematical information is encoded there? Since it is a homology theory over a space such as $\Sigma_g \times S^1$, I should be able to define it solely in terms of singular homology? Or can I somehow view instanton Floer cohomology in terms of de Rham cohomology?
Most textbooks require a lot of background, so I am happy with a vague answer to help me get intuition.