Here's a list of the more prominent save editors out there. If something's not right or if something should be up there but isn't, let us know!
Main Series
PKHeX (Gen 1 to 7)
The crown champion of save editors, this handles saves from every generation. Comes with in-built legality checking, mystery gift and trainer editing, among other things. Some niche values like underground flags in DPPt (Gen 4) aren't available, and you'll need to fall back on PokeSav for those.
PokeGen (Gen 4 to 5)
One of the leading save editors of its time, PokeGen focuses more on editing Pokemon data.
PokeSav (Gen 4 to 5)
Before PKHeX there was PokeGen, and before PokeGen there was PokeSav. Back in my day, UI wasn't something worth investing in! Some niche functionality like underground editing is still unique to PokeSav, though.
PKX Delta (Gen 1 to 3)
If you need to load Gen 1 to 3 data created by a legacy tool like PikaSav or A-Save, you'll have to fall back on PKX Delta.
pk2pk (Gen 3 to 6)
Allows for the migration of Pokemon from Gen 3/4/5 to 4/5/6.
PKMDS Code Library (Gen 3 to 6)
For other developers, this is a code library to help you get started. Currently supports Gen V and limited portions of Gen III / VI.
Online Pokemon Database (Gen 3 to 6)
The PokeCheck of today. Upload Pokemon from Gen 3 to Gen 6, manage them, and view their legality.
Rhydon Save Editor (Gen 1)
Save editor for RBY.
PkmGCSaveEditor (Gen 3)
Save editor for Colosseum and XD. Allows converting between main series games.
Trigger's PC (Gen 3)
A "Legit-Use" Everything Bank and Save Manager for Gen 3.
Mystery Gift Tool (Gen 3)
Nintendo Events, Wondercards, e-Trainer cards and e-Berry editor.
Mirage Island Finder (Gen 3)
For finding Mirage Island in RSE.
Feebas Fishing Spot Finder (Gen 3)
This tool allows to see the Feebas fishing spot seed in a Pokemon Ruby/Sapphire or Emerald savegame.
PPSE-DS (Gen 4)
Save Editor that runs ON the DS! Runs off Flashcarts and edits the save without the need for a PC.
Generation 5 Save Tool (Gen 5)
Multipurpose save game editor. Can edit the Entralink, Medals, Memory Link, Join Avenue, and Hidden Grotto.
Download Tournament Editor (Gen 5)
Tool to edit downloadable Pokemon World Tournaments in Gen 5.
Pokestar Pokemon Enabler (Gen 5)
Editor for B2W2 to modify Pokestar Studios Pokemon and the Hall of Fame.
PKMDS G5 PC Storage Sorter (Gen 5)
As an example of what the aforementioned code library can do, this tool sorts your entire Gen 5 PC in National Dex order.
WC6Full Creator (Gen 6)
Application to dump full Wondercard data from saves.
Pokemon Link Tool (Gen 6)
Allows dumping and injecting of Pokemon Link data for XYORAS.
Spin-off Series
Sky Editor (PMD: Various) and Sky JEditor (Java Port)
Save Editor for many Pokemon Mystery Dungeon games (not the 3DS ones though).
Pokemon Ranger Mission Injector
Supports Shadows of Almia and Guardian Signs.
Tools whose features are already present in a newer tool are listed below: