I have a definition in my notes that states a transposition is a permutation which interchanges two symbols and leaves all the others fixed. Thus, if $T$ is a transposition, then $\operatorname{sgn}(T)= -1$.
I understood transposition to be disjoint cycles of length 2. What does it mean when it says interchanges two symbols?
It means that all but two symbols are fixed under the action of the transposition. "Symbols" here means elements of the set being acted on (conventionally $\{1,2,\ldots, n\}$ for elements of $S_n$).
For example, $(12)\in S_4$ would be a transposition, because it only interchanges the symbols $1$ and $2$, while leaving the symbols $3$ and $4$ fixed. On the other hand, $(12)(34)\in S_4$ would not be a transposition, because it interchanges $1$ and $2$ but also interchanges $3$ and $4$.