First off, I don't mean to be condescending but I would like to clarify that it is actually the United Nations Security Council (UNSC). The resolutions in questions that began UN involvement in the Korean War are UNSC Resolution 82, 83, 84 and 85 which were all passed in 1950 by the council. Resolution 82 essentially was the UN condemning North Korea as the aggressor in the conflict which was a key political victory for the US as it framed further international involvement in the conflict as defending their sovereignty which gave them the moral high ground. Resolutions 83 to 85 were the UNSC voicing their disappointment that North Korea had not withdrawn despite UN request and authorized other countries to render military aid to South Korea and placed the US in command of UN forces during this mission. These resolutions are important as they are passed by the Security Council which is the only organ of the UN that can pass legally binding resolutions and hence authorize countries to take certain actions which gave the US a lot of credibility.
Why the USSR did not veto this was simply because they were not present at any of these security council meetings. The Soviet ambassador at the time was under orders from Stalin to boycott all UN meetings due to the ongoing dispute of which China- The PRC or ROC- should occupy the permanent seat on the security council. This was important at the time as the permanent five (P5) of the council have veto power which could bring any UNSC action to a grinding halt and at this point, four out of the five members were more or less aligned with each other since the ROC represented China. The USSR wanted the PRC to take the P5 seat in order to better balance power and to avoid political situations such as in the Korean War where they were the sole member expressly opposed to UN action which did not bode well for its international standing. The US used this opportunity where the Soviets were absent to pass the various UNSC Resolutions in quick succession in less than 40 days.
When the USSR did return to security council meetings they did cast their veto to inhibit further resolutions being passed in September which placed the UNSC into deadlock. The US responded by rallying the General Assembly (UNGA) together to pass the Uniting for Peace Resolution or Resolution 377A which allowed the UNGA to pass resolutions on security matters if the UNSC could not come to an agreement. This gave the US and the international community continued mandate to remain involved in Korea which effectively bypassed the Soviet Veto. The catch was that the UNGA still could not pass any legally binding resolutions so countries could still choose to ignore them. In this case, it was not an issue since there was strong political will from the international community to remain involved in the conflict but it would limit future uses of the Uniting for Peace Resolution.
Sources
From the UN.org archive:
UNSC Resolution 82: http://www.un.org/en/ga/search/view_doc.asp?symbol=S/RES/82(1950)
UNSC Resolution 83: http://www.un.org/en/ga/search/view_doc.asp?symbol=S/RES/83(1950)
UNSC Resolution 84: http://www.un.org/en/ga/search/view_doc.asp?symbol=S/RES/84(1950)
UNSC Resolution 85: http://www.un.org/en/ga/search/view_doc.asp?symbol=S/RES/85(1950)
UNGA Resolution 377A: http://www.un.org/en/sc/repertoire/otherdocs/GAres377A(v).pdf
Secondary Sources: