As of writing, Minecraft doesn't have anything "modding API," so members of the community have taken up themselves to write their own.
Bukkit is generally favored by most people who run servers, and most of the administrative, logging, protection, and other management plugins only exist for Bukkit. If one wants to run a vanilla server with mods, Bukkit would be the ideal choice.
In 2014, the Bukkit project ended and was taken over by Spigot, who provide all current Bukkit builds. At the same time Sponge started, and provides a more modern approach to Minecraft modding with full Forge support.
Because Forge lets mods add new blocks, items, and so on, most of the mods for Forge are primarily gameplay related and there are very few mods for management for Forge.
That is the main dichotomy within Minecraft, but there are also various off-shoots of those two.
Before Bukkit, Forge, or any of the above software, there was hMod — a server-only mod similar to Bukkit. Bukkit was created out of hMod by some of the developers (but not all) of hMod. Canary was created at the same time by some other developers.
First, we cannot make guarantees as to whether the experimental builds will break things because it represents code that is being worked on. Realistically, the experimental builds are safe 95% of the time, but occasionally there may be some major issue.
Using MultiMC is perhaps the easiest way to install WorldEdit. Download MultiMC for your system, extract it somewhere, and run the program.
Once the program is open, click the top right to login.
Then, click "New Instance" in the top left toolbar as seen below.
Choose any name for this profile and pick the MC version that you want.
Right click the profile and click "Edit Mods."
Choose "Install Forge" or "Install LiteLoader" — or even both, depending on the version of WorldEdit that you want to use.
Already downloaded WorldEdit for Forge or LiteLoader? Good. On the "Loader Mods" tab, click "Add" and add the version of WorldEdit that you downloaded.
Click "Close" and double click the profile to run Minecraft with WorldEdit!
Note! Some features, such as setting biomes or copying and pasting chests, may not work. This is because the version of WorldEdit you have chosen does not fully implement WorldEdit. The most capable version of WorldEdit is for Bukkit, a server mod, but that requires a server.