The end-point of the titration occurs at pH7 when you have a strong acid + strong base or a weak acid + weak base. The end point of a strong acid + weak base titration will be less than pH7. This is because the final product is slightly acidic.
For example, HCl + NH4OH -> H2O + NH4Cl
Although, there are no hydrogen and hydroxide ions left from the acid or base, the Cl- ions from NH4Cl can go to the H+ ions in the water equilibrium, and the NH4 ions can go to the OH- ions. However, since NH4OH is a weak base, there will be some molecular NH4OH so [H+] > [OH-].
A similar thing happens when you have a weak acid + strong base but the end point will be greater than pH 7.
For the shape of the graph, you have to remember that the endpoint is based on concentration but the axis is pH which is the log of concentration.
The steep portion of the graph near the endpoint occurs because you are increasing the concentration of hydroxide by adding alkali and decreasing the concentration of hydrogen ions by using them up. Whereas, at the start, you are mainly decreasing hydrogen ion concentration by reacting the 'excess' hydrogen ions with some alkali and at the end you are mainly increasing hydroxide ion concentration by adding the 'excess' hydroxide ions with some hydrogen ions.