There are two main areas where curved arrows are used. The first one is their use is resonance structures and the second is their use in demonstrating the mechanisms of organic reaction.
The big difference between these two is that in resonance structures the connectivity of atoms stays the same. This means that resonance structures represent the same entity only with different electron distribution.
In fact, even the electrons do not move in resonance structures and we are simply showing them as such to keep track and explained certain properties and reactivity of compounds.
The main implication of the fact that resonance structures represent the same molecule/ion is that you cannot break any σ bonds as this would change the connectivity of atoms, hence different molecules would form. So, when initially we said that curved arrows must start either from lone pair of electrons or a covenant bond, this statement is narrowed down for resonance structures: Curved arrows in resonance structures must start either from lone pair or π bonds.
In general, the following two rules must be followed when drawing resonance structures:
1) Do not exceed the octet on 2nd-row elements.
2) Do not break single bonds
There is a lot more about this in the following post (Resonance Structures in Organic Chemistry) so feel free to read the material and then continue to the next part.
Curved arrows in organic reaction mechanisms
Curved arrows are a formal notation to help us understand the electron flow in organic reactions. This makes it easier to keep track of the bonds forming and breaking during the reaction as well as visualizing and explain more advanced features such as the region and stereochemistry of certain reactions.
The electrons always flow from a high electron density region to a low electron density region. The molecules with a high electron density are nucleophiles – i.e. love nucleus. A molecule with a low electron density is classified as an electrophile – i.e. loves electrons.
The following is a nucleophilic addition reaction which is a very important class of organic reactions:
The arrow starting from the lone pair on the sulfur and pointing to the positively charged carbon makes a new covalent bond between them by a nucleophilic attack.
Another common important class of reactions that we can consider for learning the curved arrows is the acid-base reactions:
Here, the hydroxide ion is the base and it attacks the proton connected to the carbon. So, this curved arrow shows a bond-forming between the oxygen and the hydrogen. The second arrow indicates breaking the bond between the hydrogen and the nitrogen as otherwise, the hydrogen would have left with two bonds which is not possible.
Draw a second resonance structure for a) and b) and the expected products in reactions c) and d) according to the curved arrows: