Wald's General Relativity textbook reads
We can use the above definition of integration on manifolds to define the integral of $p-$forms on $M$ over well behaved, orientable $p-$dimensional surfaces in $M.$ First, we must define more precisely the notion of a "well behaved surface." Let $S$ be a manifold of dimension $p<n$. If $\phi: S \to M$ is $C^{\infty}$, is locally one to one - i.e., each $q \in S$ has an open neighborhood $O$ such that $\phi$ restricted to $O$ is one-to-one - and $\phi^{-1}: \phi[O] \to S$ is $C^{\infty}$, then $\phi[S]$ is said to be an immersed submanifold of $M.$ If, in addition, $\phi$ is globally one-to-one (i.e., $\phi[S]$ does not "intersect itself"), then $\phi[S]$ is said to be an embedded submanifold of $M.$
I am having trouble understanding the necessity of these requirements. What would be an example of a submanifold which is not "immersed"? That is, what would a pathological manifold not satisfying the "$\phi: S\to M$ being locally one-to-one" requirement look like?
Sadly, the terminology "an immersed submanifold" is suboptimal which leads to questions such as yours. The thing to realize is that an "immersed submanifold" in general is not a submanifold!
My preferred terminology is one of an immersion, which is a triple $$ \phi: S\to M $$ where $\phi$ is a smooth map such that the differential $d\phi_x$ is 1-1 for every $x\in S$. Equivalently, for each $x\in S$ there exists a neighborhood $U$ of $x$ in $S$ such that:
a. $\phi(U)$ is a smooth submanifold $N$ in $M$.
and
b. $\phi: U\to N$ is a diffeomorphism, i.e. a map which has a smooth inverse.
Given this, one can say that a subset $A\subset M$ is an "immersed submanifold" if there exists an immersion $\phi: S\to M$ whose image $\phi(S)$ equals $A$.
To make things worse, Wald's definition of an embedding (an embedded submanifold) is plain wrong! The correct definition is that $\phi$ is an immersion which is a homeomorphism to its image. (This includes, the 1-1 requirement, but is stronger.)
Examples of immersed submanifolds were discussed many times at MSE, for instance here.