Advaita Vedānta advocates nidi-dhyāsana or meditation on the nature of the ātman as the means for its realization. This should be preceded by śravaṇa (hearing about the ātman from the guru or preceptor) and manana (reflection leading to a clear concept and conviction). Once the nature of the ātman is comprehended, and a clear idea formed in the mind, maturing into a deep conviction, meditation can be proceeded with.
In meditation, the mind takes the form of the object presented to it. Such modifications are called vṛttis and are like waves unto water. When each succeeding wave is similar to the preceding wave, meditation results.
The object of such meditation in Advaita Vedānta is the Ātman itself, which again is identical in its essence with Brah-man, the Absolute. Brahman is akhaṇḍa, partless, since it is a homogeneous reality, nay, the only Reality that exists. So, when the mind starts meditating on Brahman which is akhaṇḍa, its vṛttis take on the form of akhaṇḍa-Brahman. Such vṛttis are called akhaṇḍākāra-vṛttis. When this meditation ripens into samādhi (absolute concentration), the vṛttis die down and akhaṇḍa-Brahman stands revealed.
See also NIDIDHYĀSANA.