Immersed manifold

It is a manifold $M$ together with a differentiable map

satisfying $df(X)\ne 0$ if and only if $X\ne 0$ (and therefore it is a local diffeomorphism). It is also said that $f$ is an immersion.

It is a weaker notion that embedded manifold.

This notion can be generalized to any codomain $N$ (also a smooth manifold) instead of ${\mathbb{R}}^n$. It is verified the canonical immersion theorem.

On the other hand, an immersed manifold does not necessarily give rise to a submanifold. We have to require that $f$ be injective. An immersed submanifold would be a subset $M$ of ${\mathbb{R}}^n$ such that it is a immersed manifold with the identity map. The differentiable structure of $M$, and in particular, its topology, has nothing to do with that of ${\mathbb{R}}^n$.

Theorem (local embedding theorem). Given an immersion $f:M⟶N$ and a point $p\in M$ there exists a neighborhood $V$ of $M$ such that the restriction $f{|}_V$ is an embedding.
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See @lee2013smooth Theorem 4.25