In simple words, ecumenism is the furthering of religious unity among the different Christian faiths.

The true meaning of "Christian unity" was echoed by Jesus when He prayed for His disciples. 


That they all may be one, as You, Father, are in Me, and I in You; that they also may be one in Us, that the world may believe that You sent Me. - John 17:21


In other words, that all Christian faith be of one mind in the Spirit of Truth. Regarding the subject of unity, the Apostle Paul said:

May the God of steadfastness and encouragement grant you to live in harmony with one another, in accordance with Christ Jesus, so that together you may with one voice glorify the God and Father of our Lord Jesus Christ. - Romans 15:5-6


Now I appeal to you, brothers and sisters, by the name of our Lord Jesus Christ, that all of you be in agreement and that there be no divisions  among you, but that you be united in the same mind and the same purpose. - 1 Corinthians 1:10


Put things in order, listen to my appeal, agree with one another, live in peace, and the God of love and peace will be with you. - 2 Corinthians 13:11