I haven't studied Christian contemplation or meditation in any detail. It would be awesome if somebody were to compare these techniques with Indian spiritual traditions like Sufi / Bhakti / Advaita Vedanta / Buddhist meditation. Below are some pointers to get started.
The classic text for Christian contemplation is The Cloud of Unknowing, written in the 14th century by an anonymous spiritual teacher. Detailed instructions about the technique may be found in The Cloud of Unknowing — Annotated by Joseph Mariconda. I haven't studied this book or its method in detail. However, I find glimpses of similarities between Christian contemplation and Eastern 'mantra-based meditation' fascinating!
Apart from 'Christian contemplation', there is also 'Christian meditation' which is different from Eastern meditations. In the East, 'contemplation' refers to thinking about a saying or a prayer, analyzing it, evaluating it, understanding it and absorbing it. In the East, 'meditation' refers to focusing on something like breath, an image or a mantra so that mind-chatter stops and thoughts cease. Thus 'Christian meditation' and 'Christian contemplation' probably correspond to 'Eastern contemplation' and 'Eastern meditation', respectively! Only somebody who is familiar with both approaches may be able to compare the two.