This role is very similar to that of the British constitutional monarch (and is modeled on that role to a great extent). This is admittedly weak on what makes Slovakia distinctive, but fleshes out a lot of semantic dribble of formalism in a more concrete way with examples and explanation that is shared by all Presidents of this type of which there are dozens in the world.
Per the Wikipedia link in the OP.
The President of Slovakia has a limited role in policy-making, as the
office is largely ceremonial. According to the constitution, the
president is the supreme representative of the state both in Slovakia
Among the President's constitutional powers are nominating and
appointing the Prime Minister, three judges of the constitutional
court and three members of the judicial council. The president can
also veto any bill or proposal by the National Council, except for
constitutional amendments. This veto can be overridden if the National
Council passes the same bill again with a majority of all members of
the Council, similar to the US system of presidential veto. The
president also acts as the commander-in-chief of the Slovak armed
The President's veto is a very weak one, even weaker than the delaying power of the House of Lords or the Canadian Senate. There are six judicial appointments the President gets to make. And the President it commander-in-chief of the armed forces (not entirely a ceremonial role) or the head of state in foreign diplomatic dealings.
So, there is room for significant defense and foreign policy power, and the President serves as a mild speed bump in the way of the parliament doing something rash and foolish.
Clearing House For Orderly Recognition Of Official Government Action
The remaining powers are in name only as they must be done on the recommendation of the Prime Minister.
Among their other constitutional duties are signing bills into the
law, appointing ministers on the recommendation of the prime minister
and appointing various other state officials: generals, professors,
judges, rectors, procurators and such. The president can grant pardon,
amnesty, commutations, and parole on the recommendation of the
minister of justice.
In this role, the President simply serves as an orderly clearing house for official actions so that it is perfectly clear what has been passed and done (because the President has signed it) and what is still being negotiated or is contingent or ultimately didn't amount to anything of legal effect (because it was never presented to the President for signing). Basically, the President serves the truly secretarial roles of the American Secretary of State or a state Secretary of State, or a legislative chief clerk or a county or city clerk (in addition to some of the policy roles of the American Secretary of State).
This also promotes transparency on the part of the National Council and insures that the President is kept abreast of the official actions of the government before they become official.
Power Flowing From Symbolic Leadership
Usually, notwithstanding the limited formal power of the President, the President has somewhat more actual power. And, the power of the post depends to a significant extent upon what the incumbent office holder makes of he post and that person's personal charisma, political instincts, attitudes, and connections. The President has power because the President knows how to talk to the right people with formal power to get them to agree with her point of view.
Simply put, the President is at the top of the organization chart above the Prime Minister and everyone else. This position in the organization, no matter how symbolic and limited in formal power, necessarily confers a lot of informal power in any organization upon the President.
The President has the mandate of winning a national popular vote to a five year term and the symbolic position of living in the official national residence (read palace). No other elected official in the country can say that.
One of the reasons to take away power from the President is to prevent controversies over policy and budget issues to tarnish the President's image so that if there is a crisis the President is the most unsullied person on the political field whose credibility and wide public support relative to parliamentary partisans can rally and unify the public, and can broker solutions to a constitutional or national crisis.
Everyone else in parliament is answerable to the district that elected them or to a particular political party that has means to enforce its wishes against the MP. In contrast, the President is pretty much the only person in the system who represents the entire country as a constituency, and who is, if not non-partisan, at least, more removed from day to day political controversies and expected by virtue of the office to take less partisan tone. Basically, in any political or constitutional or national crisis, everyone looks to the President to step up and lead and point the nation towards the best possible resolution.
The President also has more of the "bully pulpit" ability to command national attention in speeches on matters of importance that people take seriously and that influence people's beliefs and feelings than anyone else. Often this is the single most important informal power of the President. Presidential pronouncements make it into the news cycle. Donald Trump tweets, the President of Slovakia gives speeches and makes pronouncements, and the people of Slovakia discuss what the President has said in earnest.
Similarly, the President can, if the President chooses to, be something of an ombudsman for the average aggrieved person with little formal power but great informal power of persuasion (in part because only very connected people are appointed as President) and the President has more time to worry about these cases than government ministers running large bureaucracies.
Foreign Policy and Commander In Chief
In foreign policy and major decisions on going to war, dealing with an insurgency if there is one, or deploying troops short of war, while the President doesn't have unlimited power (and must answer to the Prime Minister for a budget and many key appointments and any needed legislation), it is also the case that the Prime Minister will have a very difficult time successfully conducting diplomacy and war and dealing with insurgencies, if any, without the willing cooperation and support of the President, which gives the President a very meaningful say in these matters.
In theory, the real hard military decisions, like authorizing forces to raid a terrorist camp, or to decide to shoot down an enemy fighter that passes into their airspace, or to decide to push forward in a conflict or fall back and regroup, are the President's decisions to make, especially in a small country in the age of telecommunications, where bringing a decision to the top in real time is often feasible, although, of course, the Prime Minister and cabinet certainly have a strong say and lots of input and their support is mutually needed. In a country like Slovakia, the President will know every top officer in the military personally. And, when push comes to shove on an operational decision involving the use of military force, the President and not the Prime Minister, is supposed to have the deciding vote when the discussions are over and is supposed to be able to decide when talking will end and the time has come to make a decision.
A President inclined to do so, of course, can be lazy and can easily punt on military decisions by deferring to the highest ranking military officials, the Defense Minister and the Prime Minister as the President does in other areas.
A Corporate Analogy
An analogy from the corporate world is to the President as the Chair of the Board, and the Prime Minister as the CEO. The President is supposed to ensure a smooth succession, to be able to smooth things over to get people to cooperate within the system, and to keep an eye on the big, long term picture. The CEO is in charge of running the business itself.
Guardian Of The Big Picture Issues For Nation's Long Term Survival
The President is supposed to have an eye on potential future crises that threaten the very existence and functioning of Slovakia as a nation-state.
If she fears imminent hostile action from Russia, she needs to sound the alarm with her own people, rally supporters on the international scene, and push to have the military handle the early skirmishes in a manner that heads off later battles and wars.
If she fears that the nation is being torn apart between religious factions or political factions or what have you, sometimes erupting into violence, she is expected to identify and discuss the problem, to preach to the people and politicians regarding their moral obligations and civic responsibilities, and to actively look for ways to build up support and consensus for solutions to problems that may takes decades to really resolve.
In case of internal division, imagine the President in a role for Slovakia similar to that of Roman Emperor Constantine organizing the Council of Nicaea to prevent a schism of the Roman Catholic church that is the new leading religion of his empire, between factions formally divided over matters of doctrine like the nature of Christ's divinity and unofficially divided ethnically and by social class and politics as well, so that united, they can try to build a stronger church and a stronger nation. The Emperor didn't have a vote in the Council or even a voice, but did help convene the people who needed to be at the table to get the institution's divisions resolved once and for all and worked behind the scenes (sometimes through representatives) to make sure that the negotiations and discussions proceeded towards a resolution that everyone would accept.
Not Everyone Will Live Up To The Job's Potential
Of course, sometimes, the President is basically a potted plant who does what he or she is told, attends lots of official funerals, state dinners and openings of new public buildings, ignores the long term threats facing the country, gets a little too embroiled in minor political disputes to stay unsullied in the public eye, and is more like the American Vice President.
If the President is a less impressive person, the President's power will be reduced accordingly, and the nation will be none the worse for it so long as the Prime Minister does a good job and there is no major crisis for the nation that needs to be addressed by the President and no one else.
Not Every President Will Have A Pure Heart
It isn't unprecedented for a President in a system like this to devote most of his or her energy to personal pleasure through corrupt schemes.
It also isn't unprecedented for a President in a system like this to become a demagogue rather than the conscience of the nation.
Slovakia Needn't Worry
Fortunately, newly elected Zuzana Caputova isn't likely to become either of these things. She is a social justice warrior in her role as a lawyer, fighting corruption and political violence when the nation is shaken by those things, and pushing for a humanistic liberal positive direction for her country at a time when the public has grown cynical about the ruling center-left party (whose last leader had to step down due to controversies that had tens of thousands of people marching in the streets). She isn't a career politician and has almost no formal political experience, and the 58.4% of the second round vote she won (against a senior EU bureaucrat) gives her a strong mandate as someone who speaks for the people.
Lots of people see newly elected Democratic member of Congress and progressive Alexandria Ocasio-Cortez in the United States as an articulate breath of fresh air and the public image of Zuzana Caputova is quite similar (albeit 16 years older).
If anyone will not be corrupt, will not be a demagogue and will not be a potted plant, which are the main worst case scenarios for a Presidency like the one that exists in Slovakia, it is her.
She may not be schooled in diplomacy and military affairs, and thus might have to defer to others somewhat in this areas, and isn't used to playing the game of movement politics from an establishment position. And, she may not have the political connections that can be useful in a post like this one. But what she does have are a sparkling clean reputation and pure heart in a nation that has grown cynical about the political leaders to whom it has entrusted formal power.