The title president has been carried over from a time when such person actually presided over (sat in front of) the government body, as with the US President of the Continental Congress, before the executive function was split into a separate branch of government. After this split, the President was no longer needed to sit in front of the legislative body, although the executive title remained in legacy.
Although not exclusive to republics, and applied in the case of semi-constitutional monarchies where a monarch exercises power (both as head of state and chief of the executive branch of government) alongside a legislature, the term is often associated with republican systems in the Americas. Presidential systems are numerous and diverse, but the following are generally true of most such governments:
The executive branch does not propose bills. However, they may have the power to veto acts of the legislature and, in turn, a supermajority of legislators may act to override the veto. This practice is generally derived from the British tradition of royal assent in which an act of parliament cannot come into effect without the assent of the monarch.
In the case of presidential republics, the president has a fixed term of office. Elections are held at scheduled times and cannot be triggered by a vote of confidence or other such parliamentary procedures. Although in some countries, there is an exception to this rule, which provides for the removal of a president who is found to have broken a law.
The executive branch is unipersonal.