- To a first approximation, there are only ragas; you don't have to care about raginis.
- If you really want to know what raginis are, then (in the sense that you are most likely to encounter the term) primarily they are (female) personifications [as counterparts] of certain ragas, used in artistic (i.e. in paintings) depictions.
- If you really want to consider ragas and raginis in the same category, then there exists an old (historical) categorization wherein, of (what are now called) ragas, some were categorized as (male) ragas, some as (female) raginis.
(I will try to edit and update this answer with more detail later. For now…)
Recall that a rāga is not just a “scale” or a “mode” but really something like: something that puts the listener in a certain mood. There is a set of notes yes, but there may also be allowed (or common) sequences: for example, notes A, B, C may all be present in a raga, but sequences “AC”, “ABAC”, may occur and “BC” never occur. Similarly, there may be certain notes used when “ascending” and a different set when “descending”. Moreover, the raga may be associated with a certain time of day, a certain season, etc.
(The “feeling” is really more important than any technical definition. An anecdote: my father is a great lover and listener of (Indian classical) music, but has no musical training. He cannot identify a single note, but he can identify many ragas in seconds (after just a few notes), even before many experts who do recognize the notes have had enough information to narrow down the possibilities consistent with the notes heard so far. The reason he can do this, AFAICT, is that the musician too is trying to be consonant with a certain “feeling”, that an experienced listener unencumbered by technical detail can grasp…)
Ragini (sense 1)
As every raga has a certain “character” and “personality”, it is only natural that someone must have tried to represent them, and they became a popular subject of paintings for a few centuries in Indian art. Certain ragas were represented as women with that character, etc.
See Wikipedia (current version) on Raga-ragini system, Ragamala paintings.
(TODO: add some examples to illustrate. E.g. a raga with a feeling of “longing” may be depicted as a woman (ragini) who is longing for something, etc.)
Ragini (sense 2)
(It is arguable whether this is different from the above or I'm making a distinction that doesn't really exist, but anyway…)
Although an astronomical number of ragas are possible in principle, in practice after you have about 100 or so (or maybe even less), some of them begin to be “similar” to others. This gives the idea of “family relationships” between the “ragas”: various music theorists have created categorizations where some ragas are male, they have female raginis as wives, together they have other ragas as children, etc.
See this article for a bit more detail and examples.
Of course, to reiterate what I said earlier, in practice if your interest is mainly in music (and not music history, or art related to music, etc.), then you can forget about “ragini” as a term. For example in a list of raginis you may find Bhairavi, Devagandhari, Dhanasri, Todi, etc., but to most musicians and listeners, these are simply ragas just like the supposedly “male” ragas like Bhairav and Malkauns; in practice no one calls any of them raginis.
(Disclaimer: I am not an expert.)