In "Introduction to Mechanics and Symmetry" by Marsden and Raţiu it is written, in chapter 10, page 347, example b, that "[s]ymplectic leaves need not be submanifolds".
In "Lectures on Poisson Geometry" by Rui Loja and Ioan Mărcuţ it is written, at page 46, Definition 3.21, that a "symplectic leaf [...] is a maximal (relative to inclusion) integral submanifold of" a certain distribution.
I'm confused: are symplectic leaves submanifolds, or not?
This is an unfortunate clash of terminology that has little to do with symplectic stuff and more to do with foliations.
Some authors require their submanifolds to be embedded submanifolds, so the irrational line on the torus is not a submanifold to them. This is the nomenclature in the first sentence. Some only require them to be immersed (so, locally embedded, but not globally). Leaves of a foliation are immersed submanifolds. This is what's meant in the second sentence, and any place you see "integral submanifold" (integral submanifolds are rarely embedded). Same example: irrational line on the torus is an integral manifold.
They're actually slightly better than just being a general immersed submanifold: they're weakly embedded, meaning that any smooth map $M \to N$ whose image lies in the immersed submanifold $S$ is smooth as a map $M \to S$. (The hard part here is that these maps are automatically continuous; after that, smoothness comes for free.)