API stands for “Application Programming Interface”. A mod API gives a programmer the ability to hook their own code into the original program without having to modify (or even really know much about) the original program’s code. Web browser plugins are a good example.
Right now, if you want to change minecraft’s behavior or add new features, you have to find the minecraft.jar file (which is a standard way to contain a java program, kind of like a zip file), unstuff the whole thing, de-compile the entire program, find the source code that you want to alter, make your own copies of it, then use your new files to build a new jar file that will basically replace the original versions.
The whole process is rather janky, even though people have made a minecraft mod toolkit it still has a lot of pitfalls and gotchas. If Mojang changes the code your mod relies on, it’ll break. If the user installs another mod that touches the same code, it might break or it might just behave very strangely.
One of the things I want to do in minecraft is add some new blocks based loosely on old school lego bricks. Things like a 45 degree up angle and down angle brick, arch bricks, etc. Basically, the really simple bricks you’d find in a Lego kit circa 1978. This will require both making the new blocks and possibly changing the behavior of current blocks depending on how they interact with the new blocks.