This second round meant that the winner would be nearly automatically ensured to be able to get a majority or at least to build a coalition with only one other party. Indeed, the winner gets a bonus of 50 seats in the Assembly of 300 representatives, usually enough to make the difference. Here are the final results in English.
In this case, and while the winner, conservative Nea Democratia (ND), did clinch a relative majority, even these 50 seats were not enough and it will have to build a coalition (129 seats < 151). Nothing too bad, mind you, as the only other party which clearly announced it could work with ND was "socialist" PASOK who got enough seats to complete a solid majority (33 seats + 129 = 162 > 151). The self declared opposition party, SYRIZA, a coalition of no-communists, socialists and anarchy-unionists, got 71 seats. But the legitimacy of such a majority is extremely low. If one looks at the percentage of expressed votes, the 162 seats represent in fact 41.94% of the voices. SYRIZA would have an easy claim that, allied with parties opposed to the ND and PASOK policy, they represent more of the people. Legitimacy is a tricky thing in a democracy. As my old Constitutional Law teacher used to say, ultimate power is not in the majority in an election but in the street violence that the people can unleash if they don't feel represented.
So the real strength of ND will be its ability to attract the support (at least in the Parliament if not in the government) of another ally, in order to get this badly lacking legitimacy. ND and PASOK asked SYRIZA to join, but its leader, Alexis Tsipras, is too clever for Greece's own good and refused flatly. He knows that his popularity is only linked to his new boyish face and empty promises. If he joins the majority, he will disappear politically. So Antonis Samaras and Evangelos Venizelos (the ND and PASOK leaders respectively) will have to cut a deal with another party or two. The most obvious candidate is the smaller Democratiki Aristera (Democratic Left), a moderate party of progressive leaning technocrats. Its leader, Fotis Kouvelis, previously rejected such a deal, but concessions by ND and PASOK and opening by German negotiators could make him change his mind. Same is true for Anexartitoi Ellines (Independant Greeks), an ultra-conservative group of Church and Fatherland types.
None of these two parties got enough votes to help the future government cross the 50% votes legitimacy threshold. But, if both join, this threshold would be passed. Such a scenario is highly unlikely. To be honest, the presence of any of the two is very likely to prevent the other from joining as their ideologies are diametrically opposed. There is only so much you can ask for the good of the country...