In terms of International Relations, the term "liberal" applies to people who believe in the importance of international institutions and contacts. They believe these are more important than the power politics that the realists focus on.
Ikenberry's argument about the post-Cold War order is based largely on liberal assumptions. According to him, the rich liberal democracies had an agreement of sorts that runs this era of our history. He says that things like free trade and the United Nations and all the sorts of international ruling bodies (even down to small one like ones on fisheries) make for stability.
The liberal democracies have all these contacts that make them peaceful and prosperous. Other countries want in on the peace and prosperity so they go along with this order. Therefore, there is no strategic global rivalry.
It will be interesting to see how the rise of China affects this analysis.