I wont touch on the powers of second generation death knights whose powers usually stem from their runeblades. However, for third gens:
Despite Decay and Death occupying different places on that popularly used image, their function appears to be exactly the same, and their names are often flung about interchangeably. They are used by warlocks, cultists, dark shamans, and of course necromancers, as well as others. This form of magic is sourced and drawn from the Shadowlands.
They, or one of the two, are responsible for one form of necromancy, in which a soul is bound to a body through death/decay, which doubles as a bit of a dark buffer between them which causes an undead to be less able to feel positive emotions (which is why sentient undead hit by the light are not just pained, but are able to be a bit happier for a while) as well as granting powerful undead the death-specific abilities such as death grip, strangulate, access to death gates into and through the shadowlands, etc.
These abilities are shown with different animations throughout Death Knight quests, sometimes they are used by hand, and sometimes through a weapon, so I’m not sure if a rune weapon is required for a DK to use these, but players certainly don’t need one. Generally, since this power comes from the energy that keeps them alive, I assume they don’t require one, especially since these abilities don’t require any kind of runes. I know much of that assumption draws on game mechanics, but I think it’s accurate, and haven’t seen anything to the contrary except for a singular instance of strangulate being channeled through an axe by one of the horsemen in the Legion campaign.
Death Knights, as you probably know, make use further of three more kinds of magic. Blood, unholy, and frost. As far as I’m aware, they don’t actually use shadow magic as a class (exceptions are of course allowed, I’m sure some of them know how to use the void), but rather magic that looks like it. This is usually unholy or the death magic I’ve written about above.
The frost magic of third generation death knights is a power that is sort of understood, but the method through which the runes function is not. The method for using them, however, is of course runeforging their runes upon their weapons, specifically frost runes, and invoking these frost runes upon their weapons (Although there is no reason these runes can’t be elsewhere, including upon armour, stitched into cloaks and tattooed upon their bodies, according to one of the unholy artifact weapon’s traits, but their weapons is the go-to location.) It should be noted that the runes of Death Knights (third generation) are stated to originate from the Northrend Vrykul, and that nobody outside of those Vrykul, including the Scourge and Ebon Blade itself, understand how they actually work. However, as we know, DKs do know how to craft and use them.
The frost magic of Death Knights, given the magic that those Vrykul use, is most likely the same kind of frost magic that the undead typically use, which is to say, death magic employed to lower temperatures, as you can see done commonly by Liches. Powerful Death Knights (and of course Liches) are able to use a more powerful form of this magic called Shadowfrost (not to be confused with Shadowfrost, which is a new frost which uses the same method, but is done with void magic instead of death magic. This is never explained outside of the tooltip for a velf racial, and is highly irritating.)
The shadowfrost of the Ebon Blade and Scourge is said to be similar to regular frost magic, however it is more powerful and able to suck the life from it’s victims. As if being frozen wasn’t bad enough.
The unholy magic DKs used was previously stated to be a merging of Death and Fel which was first invented by the Nathrezim (Dreadlords, the big demons with wings.) thousands of years ago and employed by the Scourge after it’s creation at the Legion’s hand. I think this may or may not have been retconned at some point however, I don’t remember. It does make sense, though, as the vampiric runeblades forged by the Nathrezim and given to 2nd generation Death Knights empowered them with unholy abilities. It should be noted that both Fel and Death have been shown to be able to raise the undead as skeletons and beyond, so it is not surprising that this form of magic is used by the necromancy inclined DKs, and is particularly toxic to the living and the origin of many nasty diseases transmitted through all kinds of methods, including everything from airborne miasmas, soil-infesting terraforming cauldrons converting entire regions into nightmarish horror zones, and even bug stings.
The Blood magic used by Death Knights is the magic that from what I’ve read, is the least understood. I have seen many theorise that it’s origins stem from the San’layn, elves raised by the Scourge and created with inherent hemomancy abilities, as you’ve probably seen if you’ve farmed ICC for those sweet sweet transmogs. I don’t know too much about it beyond that, except for hemomancy being used by many other groups across Azeroth. It is not surprising, however, that the highly durable Death Knights, encased in saronite armour and absolutely untouched by fatigue, mental deterioration through shadow or often pain itself, would choose this method of combat. I’d talk about it more, but their gameplay speaks for itself.
I hope that helps, I’m sure I’ve forgotten some things that I’ll curse myself about later, as usual, but that’s my understanding of the DK magics after hours and hours of reading Wowpedia and maining a DK for RP between WOTLK and the middle of MOP.
Sorry for the wall of text, I tend to ramble on and sometimes press enter to create the illusion of structure through paragraphs, heh.
Edit: And of course I was right. I forgot to mention, regarding Death/Decay, that it’s source is the shadowlands. I’ve added that tidbit to the second paragraph. You can read about the shadowlands on wowpedia quite easily. I too, cannot use links, I don’t think.