Coordination compounds with alkene ligands I came across this compound: $\ce{[Pd(η^2-C2H4)Cl2(H2O)]}$

What does that '$\eta^2$' sign mean? I am guessing that it means that the ligand is bidentate because if it wasn't, I don't think it is possible (well it would be rare) for a metal to be only co-ordinates by 3 ligands.

If this is the case, how can the carbon atom in ethene bond to Pd as it doesn't have a lone pair? Does the double break and hence the 2 carbons bond to the metal?
Ethylene is a monodentate ligand and there are 4 ligands total: ethylene,  two chlorides, and water.

Regarding hapticity, you can find a great answer here: What does the eta notation mean in the naming of a transition metal complex?

While a chapter can be dedicated to the nature of organometallic bonding, I think it is practical to say that it is the two electrons in the pi-bond that are involved in bonding to the metal center.