How does a nondeterministic Turing machine work?
Here are several ways of thinking about non-determinism (copied from this answer).

The genie. Whenever the machine has a choice, a genie tells it which way to go. If the input is in the language, then the genie can direct the machine in such a way that it eventually accepts. Conversely, if the input is not in the language, whatever the genie tells the machine to do, it will always reject.

Hints. The machine computes a bivariate function. The first input is a word $w$, and the second input is a "hint" $x$. Whenever the machine faces a non-deterministic choice, it consults the next hint symbol, and operates accordingly. We are promised the following:


Completeness: if $w \in L$ then there is some hint $x$ which causes the machine to accept.
Soundness: if $w \notin L$ then the machine rejects on all hints.


Accepting computations. An accepting computation is a legal computation (one in which the machine always operates according to one of the choices it is faced with) which ends at an accepting state. A word is in the language iff it has an accepting computation.

We can formalize the notion of accepting computation using configurations. A configuration is an instantaneous description of the entire state of the machine. We can define a relation $\sigma \vdash \sigma'$, where $\sigma,\sigma'$ are configurations, which holds when $\sigma$ can lead to $\sigma'$ in one step. In a deterministic machine, there is at most one $\sigma'$ per each $\sigma$, whereas in a nondeterministic machine, there could be more than one. An accepting computation for a word $w$ is one which starts at the initial configuration (the tape contains $w$, the head points at the start of $w$, the state is the initial state) and ends at an accepting configuration.

Another equivalent description is in terms of reachability. Consider a directed graph in which vertices are configurations and there is an edge from $\sigma$ to $\sigma'$ if $\sigma \vdash \sigma'$. An accepting computation is a path from the initial configuration to an accepting configuration.