Q:

Why do presidential pardons exist in a country having a clear separation of powers?

This answer provides some insight into a way for a President to pardon someone with virtually no consequences:

(..) in reality, presidents often wait until his final days/hours to
  grant the most controversial pardons.
And once a pardon is granted, there is no recall of the pardon.
Therefore, there is really no checks to pardon power, making it the
  most KING like power the presidency holds.

I am wondering why do modern democracies (with rather complex check and balances in place) allow pardons to exist in the first place. In a political system having separation of powers and where persons can appeal if they think the verdict is not correct, it does not make much sense to have the president pardon someone. 
Question: Why do presidential pardons exist in a country having a clear separation of powers?
Note: I will narrow this to US only, since there might hard to provide an answer to cover all countries that have presidential pardons. 

A:

There is a proverb that hard cases make bad laws. The pardon power is a check and balance on laws and their application. 

The legislative has to write laws with general scope. 
The judiciary interprets those laws in specific cases, and their role in questioning laws is rather limited. They can only set the law aside if they can find overriding principles in a constitution or the like to justify why they are ignoring the law as written.

The executive may be given the power to pardon whenever the application of a generally just law to a specific case creates an injustice. 

A:

Question: Why do presidential pardons exist in a country having a
  clear separation of powers?

In a word: history...
The case of the US is tied to that of the UK. In the UK you have the Royal Prerogative of mercy which has been used since at least 1617.
This is likely copied from across the Channel in France, where the Droit de grâce (Fr) was used by Philippe VI to pardon nobles and then got extended to the rest of the population by John II in the 14th century. The power still exists today and lies in the hands of the French President.
The point in the above historical note is that the Head of State's right to pardon was likely seen as a matter of course when the US constitution was written. Article II, Section 2 almost certainly reflected this line of thought (see Federalist 69 mentioned in C'est Moi's answer), and the POTUS right to pardon has been a thing since -- with some haggling in between with some SCOTUS cases over how it should get exercised.
Interestingly, the prerogative evolved similarly on both sides of the Atlantic. Namely, on the British side of the Atlantic it seems to have been taken out of the monarch's hands in practice, and it is the PM who suggests a pardon. On the American side of the Atlantic, the Department of Justice has an Office of the Pardon Attorney to review cases. But in both cases the power still formally remains in the hands of the Head of State.
In some sense, what's unusual with the current US administration is that Trump appears to be exercising the power on a whim without bothering to consult the Office of the Pardon Attorney, and in the middle of his presidency at that. By contrast, and as you note, his predecessors usually stuck to the very last days of their presidencies to issue controversial pardons. (Nixon's pardon by Ford is the only major exception that springs to mind in recent memory. That one was done in Ford's early days in office, in an effort to try to heal the nation.)

A:

The President has pardon power in part to prevent abuses by the other branches of government
Let's start with this statement of yours:

In a political system having separation of powers and where persons can appeal if they think the verdict is not correct, it does not make much sense to have the president pardon someone.

Assume for the moment that you are a wrongfully convicted person, or someone who is correctly convicted of something that shouldn't even be a crime. Who do you appeal to who? You appeal to higher courts, typically. What happens when you've run out of higher courts to appeal to? 
Suppose that the courts have decided against you every time because of some inherent flaw in the law, either as it was written or as a judge interpreted it. If you need a concrete example, imagine that you are black person living in 1850's who has escaped slavery (e.g. Dred Scott - see Dred Scott v. Sanford)
If your problem is with the law as it's written, then the legislature would have to re-write it, but that wouldn't help you; just other people like you in the future. Maybe the court could help you by vacating the law,  but if your problem is with the judicial system deciding that you have no rights at all, then that's not going to happen.
Dred Scott was "lucky" in the sense that he was privately manumitted by a new owner after the Supreme Court decided he had no rights whatsoever. What if you, as an individual, were not so "lucky"?
What you need is a third branch of government who can check the abuse that you are suffering at the hands of the legislature and the judiciary. That would be the executive, because that's the only one that's left. By pardoning you, the President can sidestep the entire question of whether or  not the law is bad and simply say that you shouldn't be in prison/facing death whatever.
(It's a little more thorny as to whether or not a pardon would have actually helped Dred Scott; I just picked it because it was an obvious injustice that was nonetheless sanctioned by the judiciary at the highest level).
There's probably a bunch of common law historical context about how the President gets to do it because Kings used to do it, but that makes things less obvious rather than more obvious about how the pardon power is still important when powers are separated.

