Why we do have different day of Eid around the world. I have seen mostly in our country we have Eid day one day after Saudi Arabia. what is the reason for that?
The Islamic calendar is a lunar calendar where each month starts with the sighting of the new moon. As some Muslim scholars adopted calculation methods (and they are not in consensus) to calculate the "new moon" and others still relay on the physical moon-sighting they might have differences. For example if the sky is full of clouds you won't see the moon in the (supposed) last day of a month. Also some scholars allow sightings which are strictly speaking not correct. A day according Islam starts with the sunset (not at midnight) so a correct moon sighting must be around this time if the new moon has been seen later that would strictly speaking count for the following day. But some allow it as long as the moon has been seen before sunrise!
And as stated in the comment of goldPseudo there's an other point of dispute which is the view of some scholars to unify the Ummah by a global sighting, even there there are different views some say that the possibility of a view anywhere on the globe should be significant others say that as Mekka is our Qiblah the view possibility in Mekka counts. As these possibilities are based on calculation there are differences in the used calculation methods!
Note that this unification would mean that those people who live in a country where the moon could be sighted at the calculated day would see a crescent moon on the first day of each month, while those whom might live elsewhere would need a day or two to be able to see it as a one or two days old moon... so they might have a feeling that the months didn't start at their location until they might be able to see the moon!
Therefore some countries still relay on the local sighting however last May there has been a conference with the goal of a global sighting in Istanbul as you may read in the start page of moonsighting.com.
Note actually the view of almost all classical scholars is that relying on calculation for the purpose of determining an Islamic date is not permissible. This actually is the official view of the scholars of Saudi Arabia however you would hardly find them fasting in a difference to the doubtful calculations algorithm of the so called Umm al-Qura calendar.
I also recommend you to take a look at the following posts:
Voluntary fasts just after Ramadan
Figuring out right calculation method of prayer time
Hijri vs English Calendar. How old is Islam?
Why does the Hijri calendar continue indefinitely?
Is there any evidence for setting the Islamic calendar through calculations only?