Śrīla Rūpa Goswami has analyzed the various manifestations of the Lord in the Laghu-bhāgavatāmṛta. He writes:
स्वयम्-रूपस् तदेकात्मरूप आवेश-नामकः
इत्यसौ त्रि-विधं भाति प्रपञ्चातित-धामसु
In His transcendental abodes, the Lord appears in three different ways, namely, as svayaṁ-rūpa, tad-ekātma-rūpa and āveśa (LBH1.11)
Svayaṁ-rūpa is the original form of Śrī Kṛṣṇa, which is ‘medium-sized’ . This form is the original form of Bhagavān, and is completely independent of any other form. That is, this form is not an ‘expansion’ from any other form. Furthermore, Śrī Kṛṣṇa is present in this svayaṁ-rūpa as Śyāmasundara everywhere in space. That is, he pervades the whole of existence in his medium-sized form. This is of course inconceivable because a form which is limited in size cannot be present everywhere. However, this has to be accepted because it is the direct experience of Śrī Nārada.
In Anuccheda 42 of the Bhagavat Sandarbha, Śrī Jīva Goswami analyzes Śrī Nārada’s experience of how Śrī Kṛṣṇa married 16,000 wives simultaneously in 16,000 different palaces. Each of these forms was performing different acts. And yet, these are not different persons, but the same person, Kṛṣṇa manifesting (prakāśa) in different places at once. This is possible for Kṛṣṇa because He is not limited by time and space. Such forms are called prakāśa forms. This is also defined in the Laghu-bhāgavatāmṛta:
अनेकत्र प्रकटता रूपस्यैकस्य यैकदा।
सर्वथा तत्स्वरूपैव स प्रकाश इतीर्यते।।
The appearance of a single form in many places at the same time, but which is, in all respects, the same essential form of the Lord, is called prakāśa.
Kṛṣṇa’s dancing in the Rāsa dance simultaneously with different gopīs is another example of prakāśa manifestations. Prakāśa forms are not ‘expansions’ of Kṛṣṇa, but rather Kṛṣṇa Himself.
Kṛṣṇa sometimes can manifest a four-handed form, but this form is not considered different from the svayaṁ-rūpa.
Forms that appear different from Śrī Kṛṣṇa’s form and have different pastimes from Him, like Nārāyaṇa, or Viṣṇu, but which are non-different in nature from Kṛṣṇa (being the same person), are called tad-ekātma-rūpa forms. These forms are further divided into vilāsa forms, such as Nārāyaṇa, and svāṁśa forms, such as Matsya and Kūrma.
āveśa forms are empowered living entities. Kṛṣṇa empowers such beings with specific potencies, such as knowledge or bhakti or strength. These persons are called āveśa-rūpa manifestations. Examples of this are Śesa, Nārada, Sanaka and others.