The president is a titular head in these type of governments.
Does the president possess power, if any at all?
If he doesn't, why keep the president? Why not just fire him/her?
I'm specifically interested in Bangladesh.
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So there are two systems of government where a President exists: Presidential System Governments and Semi-Presidential System Governments.
In the former, the President is both the head of state and the head of government. Usually these forms of Governments are distinguished in that the Executive Duties of Government are separated from the law makers (legislature). This is the most popular form of Government in the Americas, as only four contiguous Nations are not Presidential-Systems.
Across the Pond in Europe, if a nation has a President, it's normally a Semi-Presidential system. The definition of this is a little nebulous, but usually they are hybrids of Parliamentary systems and Presidential Systems. In both forms of Semi-Presidential Systems, the President appoints the Prime Minister and the Cabinet, with support of the Parliament.
The major difference between these systems is who can dismiss the Prime Minister and Cabinet. In Presidential-Parliamentary Systems, the PM is answerable to both the Parliament and the President. The President may dismiss the Prime Minister or Cabinet and Parliament can vote no confidence. The President can also dismiss parliament as a whole, which forces the step down.
In Presidential-Premier systems, the President names the Prime Minister and the Cabinet, but only Parliament may remove them. Some systems do give the President the authority dismiss parliament as a whole, but the point here is the President can not directly remove the Prime Minister.
Beyond that, the duties depend on which office handles which duties... In Bangladesh, the President is ceremonial and more akin to the Monarch in a Constitutional Monarchy (Presidential-Premier system), but in Russia (Presidential-Parliamentary), the President normally handles matters related to foreign affairs of the nation and the PM handles the domestic and internal affairs.
Depends on the country.