Well to start with, there were conflicts ongoing during the Peace Conference. The most prominent would probably in the Middle East, where the Arabs were angling to establish a kingdom in modern Syria and Jordan, despite the presence of the British and French in the region; Turkey, where the Ottoman Empire was falling apart and Allied powers were trying to establish control; and Poland, who was fighting the Bolshevik Red Army. I'm more familiar with the latter, so I'll touch on that.

Poland had not existed as an independent state for 123 years, being divided between Prussia (later Germany), Russia, and Austria. However on November 11, 1918 they declared independence, and immediately began to fight to assert control over what was considered Polish territories. Small conflicts with neighbouring new states like Czechoslovakia and Ukraine saw some borders defined, but the biggest concern was the Bolsheviks, who had taken control of Russia in November 1917.

The Bolsheviks at this time were expansionist, and had sights on invading Central Europe and helping start a revolution there, especially in Germany, which having just lost the war was seen as ripe for a socialist revolution. The Russian Civil War muted this plan for a while, but by early 1919 the Red Army was starting to look west. However in their way was the equally-expansionist Poles.

Poland was led by Józef Piłsudski, a general and socialist himself, who wanted to establish a sort of federal union of peoples in the region (Poles, Lithuanians, Belarusians, Ukrainians) under Polish hegemony, a sort of successor to the Polish-Lithuanian Commonwealth of centuries past. This would mean Poland ruling over millions of non-ethnic Poles, and would also serve as the first line of defence against a possible Bolshevik invasion (Piłsudski may have been a socialist, but he was not a Bolshevik). Conflict broke out in Ukraine as both sides fought to control the region, with Kiev changing hands something like 8 times during this period (I don't have the exact number on hand, but it was considerable).

To alleviate pressure in Ukraine the Bolsheviks launched an invasion of Poland proper in 1920, reaching the outskirts of Warsaw before being decisively defeated in the Battle of Warsaw (also known as the Miracle on the Vistula, for the unexpected victory by the Poles). Forced back into Russia, the Bolsheviks began negotiating a peace treaty, and in March 1921 the Treaty of Riga was signed, which gave Poland control of much of modern Belarus, western Ukraine, and part of Lithuania. These terms did not please anyone, as the Bolsheviks lost a lot of land that had been part of Russia, while the Poles didn't get everything they wanted, and kept tensions between the two high until 1939, when the Soviets again invaded (and promptly annexed the lost territory).