Radharani is Krishna’s Eternal Consort and Pleasure Potency. Radharani is actually Krishna Himself who have expanded himself. They both live in Supreme Spiritual Abode ‘Goloka Vrindavana’. Besides Protection of dharma,They both appear in this material world to taste different types of rasa.
Quoting from this website:
Radharani, Pleasure Potency of Krishna
His Divine Grace A.C. Bhaktivedanta Swami Prabhupada Aacharya of Brahma Sampradaya says —
Festival Lectures, London, 29 August, 1971.
So today, this evening, we are talking about Radhastami. We are trying to understand the chief potency of Krishna. Radharani is the pleasure potency of Krishna. As we understand from Vedic literature, Krishna has many varieties of potencies.
Parasya shaktir vividhaiva shruyate [Chaitanya-charitamrita Madhya-lila 13.65, purport].
Just like the same example, as a big man has got many assistants and secretaries so that he hasn't got to do anything personally, simply by his will everything is done, similarly, the Supreme Personality of Godhead has got varieties of energies, and everything is being done so nicely. Just like this material energy. This material world, where we are now living… This is called material energy.
Bahir-anga-shakti. The Sanskrit name is
bahir-anga, external energy of Krishna. So how nicely it is being done, everything in the material energy. That is also explained in the Bhagavad-gita, mayadhyakshena prakritih suyate sa-characharam: [Bhagavad-gita 9.10] "Under My superintendence the material energy is working." The material energy is not blind. It is… On the background there is Krishna. Mayadhyakshena prakritih [Bhagavad-gita 9.10]. Prakriti means this material energy.
Similarly… This is external energy. Similarly, there is another energy, which is internal energy. By the internal energy the spiritual world is being manifested. Paras tasmat tu bhavah anyah [Bhagavad-gita 8.20]. Another energy, para, superior, transcendental, the spiritual world. As this material world is being manipulated under the external energy, similarly, the spiritual world is also conducted by the internal potency. That internal potency is Radharani.
Radharani…, today is Radharani's appearance day. So we should try to understand Radharani's feature. Radharani is the pleasure potency, hladini-shakti. Anandamayo 'bhyasat (Vedanta-sutra 1.1.12). In the Vedanta-sutra the Absolute Truth is described as anandamaya, always in pleasure potency. That anandamaya potency… Just like ananda. When you want ananda, pleasure, you cannot have it alone. Alone, you cannot enjoy. When you are in the circles of friend or family or other associates, you feel pleasure. Just like I am speaking. The speaking is very pleasing when there are many persons here. I cannot speak alone here. That is not ananda. I can speak here at night, dead of night, nobody here. That is not ananda. Ananda means there must be others. So because Krishna, the Absolute Truth, is anandamaya, therefore eko bahu syam, He has become many. We are also Krishna's part and parcel, to give pleasure to Krishna. And the chief pleasure potency is Radharani.
radha-krishna-pranaya-vikritir hladini-shaktir asmad
ekatmanav api bhuvo (pura) deha-bhedo-gatau tau
chaitanyakhyam prakatam adhuna tad-dvayam chaikybram aptam
radha-bhava-(dyuti)-suvalitam naumi krishna-svarupam [Chaitanya-charitamrita Adi-lila 1.5]
So Krishna is Param Brahman, as you know from the Bhagavad-gita. When Arjuna understood Bhagavad-gita, he affirmed Krishna, param brahma param dhama pavitram paramam bhavan [Bhagavad-gita 10.12]. So Krishna is Param Brahman. So in this material world we see that a great, saintly person, simply to enjoy brahmananda, he gives up everything of material enjoyment. He becomes sannyasi. Aham brahmasmi. Just to understand that he is in Brahman realization. So if one has to give up everything material for Brahman realization, do you think that Param Brahman, the Supreme Brahman, can enjoy anything material? No. Krishna's enjoyment is nothing material. This point should be understood. For Brahman realization we are giving up everything material. And how Param Brahman can enjoy anything material? This question has been very much nicely discussed by Jiva Gosvami.
So when the Param Brahman… First of all, the Param Brahman information is not there in this material world. Little Brahman information is there. Or little Paramatma information is there. But not Param Brahman, or Bhagavan, information. Therefore it is said, manushyanam sahasreshu kashchid yatati siddhaye [Bhagavad-gita 7.3]. Siddhaye means to understand Brahman or Paramatma. But out of many such persons who have realized Brahman and Paramatma, hardly a person can know Krishna. And that… First of all…what we can understand about Krishna's pleasure potency? If I want to know some big man. That is one process. And without knowing that big man, how I can understand about his internal affairs? Similarly, if we do not understand Krishna, how we can understand how Krishna is enjoying? That is not possible. But the Gosvamis, they're giving us information what is the pleasure potency of Krishna. That is Srimati Radharani.
This answer by Be happy explains why Radha didn't marry Krishna. Quoting from that answer:
So as per BrahmaVaivarta Purana, in Goloka, before the manifestation of their pastime on earth, on one occasion Radharani had a heated argument with Shridama, the personal attendant of Krishna.
In course of their conversation Radharani got angry and cursed Shridama to take birth as a demon. So Shridama also in return cursed Radharani to be born as a human, and after spending pastime with Krishna in Gokula She will get separated for 100 years from Her beloved lover Krishna:
manuṣyā iva kopaste tasmātvaṃ mānuṣī bhubi
bhaviṣyāsi na saṃdeho mayā śaptā tvamambike [BVP - 4.3.103]
You are getting angry like a human, hence you'll become a human. Mother! you'll become a human without any doubt due to my curse.
gokule prapya taṃ kruṣṇaṃ vihariṣyasi kānane
bhavitā te varṣaśataṃ vicchedo hariṇā saha
punaḥ prapya tamīśaṃ ca golokamājamiṣyasi [BVP - 4.3.105,106]
In Gokula getting the company of Krishna you will enjoy pastimes in the forest. Then you'll experience a separation of hundred years from Hari. After that you'll again get His company and return to Go-loka.
Hence, even though Krishna's love for Radha was eternal, due to the effect of the curse He couldn't marry Radha socially or openly. Instead, as per the curse, on Earth Radha had to marry Rayana of Vrindavan who was also Kirshna's partial incarnation:
mūḍhā rāyaṇapatnīṃ tvāṃ vakṣanti jagatitale
rāyaṇaḥ śrīhareraṃśo vaiśyo vrundāvane vane [BVP - 4.3.104,105]
On Earth fools will say you to be the wife of Rayana. Rayana will be the partial incarnation of Hari in Vrindavan.
For more details, see the below posts.
How was Radhaji born? What are the names of her father and mother?
What happened to Radha when she grew up?
In the Nimbarka Sampradaya, Radha's relationship with Krishna is characterised by the svakiya-rasa, based on texts in the Brahma Vaivarta Purana and the Garga Samhita describing the wedding of Radha and Krishna. The Gaudiya tradition focuses upon parakiya-rasa as the highest form of love, wherein Radha and Krishna share thoughts even through separation. The love the gopis feel for Krishna is also described in this esoteric manner as the highest platform of spontaneous love of God, and not of a sexual nature.