Organic mechanisms are simply a description of the bond-making and bond-breaking events that occur when starting materials are converted to products. The "curved arrows" used to describe these events were pioneered by Sir Robert Robinson in the UK and are now the convention used by chemists throughout the world.* While undergraduate students often struggle with this idea it is as simple as driving on the correct side of the road; curved arrows always start at electron-rich areas and are pushed towards electron-poor areas.
If we think of electron density as a currency then the idea of "electron-rich" donating to "electron-poor" makes sense as long as we can identify sites of electron excess and electron deficiency. In Equation (i) the green curved arrow is used correctly to describe the lone pair from X (electron-rich) being used to form a covalent bond to Y (electron-poor). The green arrow in Equation 2 is incorrect because Y does not have any extra electron density to share with X. In most of the reactions we study a curved arrow will start at a lone pair or pi bond.
Since most of the reactions we will encounter involve polar mechanisms it is important to get this convention worked out early; you must be able to identify electron-rich and electron poor quickly. If you are not sure how to do this, refer to the Concepts page on the top menu. Once you become confidant with the arrow-pushing convention it can actually be used to help predict the products in new reactions.
If we start with a simple acid-base reaction such as in Equation (iii) we can identify the base and acid on the left and the conjugate acid and conjugate base on the right. Keep in mind that we are looking for patterns that will be easily recognizable as the molecules and the reactions become more complicated. Here the Na+ means that the N is going to be negative and thus electron-rich. The H atoms in the water molecule will be electron-poor since they are attached to the more electronegative O atom. This sets up the situation in Equation (iv) below.
Now we are ready to push the curved arrows but how many do we need and where do they go? Since we need to make one bond (N-H) and break one bond (O-H) we will need two curved arrows. As shown in Equation (v), the first arrow will start at N and go towards H to mimic the formation of the new N-H sigma bond; the second arrow will start at the H-O bond and go to O to avoid the H having too many electrons.
If we assess what needed to be done in the acid-base example we can extrapolate that to organic mechanisms since the number of curved arrows will match the number of bonds that need to be formed and broken. The first carbon-based mechanism usually studied in the Organic sequence is the bimolecular substitution of a nucleophile on an electrophile with loss of leaving group, which actually boils down to one bond being formed and one being broken. Considering Equation (vi) it appears that the C-CN bond must form and the C-Br bond must break.
As with the acid-base reaction we identify the electron-rich species (cyanide anion) and the electron-poor species (C in the methyl group). One bond needs to form and one needs to break so, knowing that this reaction is bimolecular and concerted, the nucleophile is going to kick out the leaving group on carbon as described in Equation (vii). This simple preliminary analysis of what needs to happen in each reaction will take a lot of the mystery out of mechanism.