Red is primary in direct damage and has it in effectively all forms, with the most straightforward and unconditional versions. The accompanying flavor and imagery of fire and electricity is responsible for the burn teminology. Red will occasionally deal damage to all or a subset of creatures, and some will also deal damage to planeswalkers in the same spell. The damage number can be any number, though the most common is the 2-4 range.
Red may also deal damage to a creature that's been damaged earlier (preying on the weak). It also may sacrifice a creature and deal damage to creature/player equal to the sacrificed creature's power/toughness. It may also deal damage to planeswalkers directly, due to the planeswalker redirection rules change.
Famous burn spells
Burn spells are most commonly used in red aggro decks, and have a low mana cost. A typical Burn Deck is a deck that seeks to reduce an opponent's life total from 20 to 0 as fast as possible, usually in the form of twenty or more instant and sorceries and a few creatures. Pure burn decks tend to suffer in the mid-game, when spells have been used up, a hand size is minimal, and there are no creatures to block with.
Black is secondary, it will do direct damage to creatures or players and then gain life (drain). Black will sometimes deal damage to players as a punishment. Black used to mostly do life loss to players, but as R&D has been trying to give black more answers to planeswalkers, they have been shifting this more into damage. With black getting direct destruction, the life-loss/damage design difference is still nebulous. Black's ending-the-game player damage tends to template as "each opponent" whereas red will more often use "target opponent/player or planeswalker".
Like red, black may deal damage to a creature that's been damaged earlier or sacrifice a creature to deal damage.
White's direct damage only shows up in combat. It is primary in dealing damage to target attacking or blocking creature, occasionally to multiple or all attackers or blockers. It also has damage to tapped creatures on the premise that such a creature probably attacked. White used to have access to damage redirection and damage based on color hate, but these are not in the current design pool.
Green may deal damage to a creature with flying. Green also gets fight and biting spells that have damage effects, albeit with a creature proxy instead of the spell. Charge of the Forever-Beast and Monstrous Onslaught have different templating that require creatures but have the spell doing the damage. There are a handful of Green direct damage spells including Hornet Sting and Unyaro Bee Sting, but these are considered mistakes and color pie breaks. Green has had some forms of player damage, but those are not considered part of the modern color pie; it has, however, started to damage planeswalkers with their bite spells.
As always, colorless cards present a way to get effects in decks that might otherwise not have access to them at rates overcoated relative to colors that are normally able to produce them.
- Unstable 's Super-Duper Death Ray combines direct damage with trample. Mark Rosewater tried for years to get trample on direct damage spells in black border, resulting in Liquid Fire.  The rules issues were later smoothed out and so Flame Spill was printed in Ikoria: Lair of Behemoths and Pigment Storm in Strixhaven: School of Mages.
- Aaron Forsythe (June 3, 2005). "Feel the Burn". magicthegathering.com. Wizards of the Coast.
- Mark Rosewater (June 5, 2017). "Mechanical Color Pie 2017". magicthegathering.com. Wizards of the Coast.
- Mark Rosewater (August 16, 2017). "In what order are the colors able to deal direct damage (i.e. burn)?". Blogatog. Tumblr.
- Mark Rosewater (May 10, 2014). "How did direct damage get the nickname burn?". Blogatog. Tumblr.
- Mark Rosewater (February 01, 2014). "Fire is cool and all, but I feel that this sort of thing could limit Red's repertoire, as far as flavour goes.". Blogatog. Tumblr.
- Mark Rosewater (November 15, 2010). "Feel the Burn". magicthegathering.com. Wizards of the Coast.
- Mark Rosewater. "Would you consider hornet sting appropriately in...". Blogatog. Tumblr.
- Mark Rosewater. "Is hornet sting worse than gut shot in terms of...". Blogatog. Tumblr.
- Mark Rosewater (November 16, 2017). "Thank you for Super Duper Death Ray.". Blogatog. Tumblr.
- Mark Rosewater. "Some time ago you mentioned that the rules cant...". Blogatog. Tumblr.
- Mark Rosewater. "Why not just put trample on a burn spell the...". Blogatog. Tumblr.