I was reading one of the possible definitions of semisimple modules, which is "for every submodule $N$ of $M$, there exists a submodule $P$ such that $M=N \bigoplus P$. After reading this I immediately thought that a particular case of semisimple modules are simple modules: if $M$ is simple then the only submodules are $0$ and $M$, and $M=0 \bigoplus M$, so the simple modules satisfy the conditions of the definition of semisimple modules.
However, I've also read that not all simple rings are semisimple rings, but a simple ring in particular is a simple module, so how can it be that it satisfied the definition of semisimple module but is not semisimple?
I am very confused by this, I would appreciate if someone could clear up my confusion, thanks in advance.
Be careful. A simple ring $R$ is "A ring with no proper nontrivial two-sided ideals", but a simple module is "A module with no proper nontrivial submodules". A semisimple ring $R$ is "A ring that is semisimple as an $R$-module". Every ring that's not a division ring has left ideals (and thus is not simple as a module over itself). Simple rings do not need to be simple as modules over themselves. (For instance, consider $R = M_2(\Bbb Q)$.)
You are correct that every simple module is a semisimple module. But the notion of simple rings and simple modules don't coincide.