Shri Radha is the 'Alahadini Shakti of Anant Alahad Shri Skrisna' (the source of bliss for the infinite bliss Shri Krishna). there is seemingly an inhetrent contradiction in this definition. how can one be a source of bliss for someone who is already infinite bliss? remember infinity is not a large number, it's a mathematical concept. Shri Krishna being 'infinite bliss' as Vedas define him, means that even if Shri Krishna himself is to experience himself for an infinite period of time, still he himself won't be able to say how much bliss he is,because an infinite 'part' of him which is again infinite bliss would still be left unexperienced by Him, and that's why he is 'infinite bliss' (Anant Alahad or Anant Anand as Vedanta mentions at numerous places). And Shri Radha is defined, by none other than Shri krishna himself,to be Krishna's Alahadini Shakti. "Radha, you are the source of bliss for me" Shri Krishna says. is there a contradiction here? no, there's no contradiction, as we will see in a moment.
as vedas again say about Shri Bhagwaan, "He doesn't like to be alone", again even this statement of Vedas appear contradictiory at face value, because Vedas themselves first described & defined Shri Bhagwaan as "infinite bliss who has infinite names,forms, attributes, divine plays (leela), eternal blissful abodes (dhaam), powers, attendants, devotees and saints, who is complete in Himself and nothing and no one exists other than him." So first Vedas define Lord as 'complete in himslef' then say 'he doesn't LIKE to be alone'!! As you can see Cool, if Lord is complete in Himself, HE can't 'like' or 'dislike' anything, he can't have any'wants' to 'desires' to have someone or something other than him. So are Vedas furthering a contradiction here when they say, 'He dosne't like to be alone'?? Again, no there is no contradiction, and we'll explain why there is no contradiction.
the best way to understand this is by taking a very simple example. Let's say Cool&Fresh is an extremely beautiful girl.and she is also infinitely powerful so she can do whatever she feels like doing. now the situation is, there is no one and nothing except Cool&Fresh, she is all alone, for all times, so while she knows very well that she is extremely beautiful,she also realises that this fact has no meaning because there is no one except her to notice this fact, and there never will be, because nothing exists or can ever exist except her!!
but because C&F is also infinitely powerful, she decides to manifest from her, another 'self' of her, so that that seemingly 'other' who in reality is C&F only, will notice and acknowledge C&F's existence and beauty!! now since this 'other' is another manifestation of C&F only, naturally this 'other' too will be extremely beautiful, and now C&F has an eternal partner with her, which she achieved by manifesting herself as 'two' and these 'two' can now keep telling each other just how extremely beautiful they both are.
that's all that is there to vedas proclamation that 'Shri Bhagwaan desires a second for he doesn't like to be alone' and 'Shri Radha is the alahadini shakti of Shri krishna' i.e. Shri Radha exists to provide bliss to infinite bliss Shri Krishna.
Shri krishna himself explains this very concisely, as is His wont, when he says to Shri Radha, "My dearest, you're me and I'm you, and anybody thinking of us in any other manner is an ignoramus."
P.S. 1. Whatever I've said C&F, if you desire, I can quote the same thing from ancient Sanskrit texts too, but that will require some effort on my part, unlike typing this post, for I'll have to search the old texts. So if you wish to read quotes from Sanaskrit texts, don't hesitate and tell me, I'll search and quote, but if you're content with reading my half-baked sentences, nothing better.
2. if you've read carefully what I've written, a question should naturally arise in your mind. That since the Supreme Lord is 'complete in Himself' why he ever takes the trouble of manifesting infinite creations in Him where infinite of his chaitanya powers or chaitanya ansh i.e. us, can experience their existence?? because Since Lord is 'the only one' and 'complete' (the Sanskrit terms for this are 'AtmaRam' and 'Poornakaam') he can't 'desire' or 'wish' to 'do' anything, he doesn't ever need to 'do' anything. So why wish to manifest infinite creations in Him, where we can experience our existence?? Answer to this is again closely related to Shri Radha. if interested, tell me, I'll talk about it next time.