Magic is an inherent element of the multiverse.
Divine magic is mediated access to this element. It's based on getting power from some deity-like being directly, or attuning to universal principles like those represented by cleric domains. Generally this is done through some form of faith or meditation or conviction, or having that connection directly conferred to you by some sort of ritual from someone else with that connection.
Arcane magic is direct access to this magical element. You make the magic happen in one of a couple ways.
It can be done scientifically, by learning words, gestures, and materials which, when combined with directed intent, invoke and channel this power in a more focused way than its ambient existance all around you. Ths is how wizards do it.
Or you can have an inborn (or similar) connection that gives you an intuitive feel for this magical energy, and makes it easier to learn how to channel and focus it. Typically you use the same components to direct it, but you grasp them intuitively and channel it without needing to ritually prep your brain (ie, without spell preparation). This is how sorcerers do it.
Innate magic can be considered a form of arcane, kind of like an extra innate sorcery. Or you can like at it as in-between divine and arcane, where it is somewhat mediated by the nature of your species. Or you can imagine it as more like a manifestation of the magical from its raw form, which defies the arcane/divine divide.
Warlock magic is arcane magic that mostly works like sorcery, but rather than being inborn you acquire it from another magical being. So it has a mediated acquisition, but an arcane nature. Or you can view it as borrowed arcane magic where an entity that uses arcane (rather than divine magic) sets up a funnel to channel it to you. By contrast, even deities themselves use divine magic, mediated by essential principles of their portfolios, which would distiguise this from divine magic.
Bards basically use highly flavored sorcery. They include extra techniques involving words of power not utilized by wizards and sorcerers.