I think it means TRANSPOSE, but I can't figure out the need to perform a transpose operation:
Similarly, the set of weights that go with the inputs is written with a small T as well:
There is also some algebraic operation with this T that I don't understand (why is the T interchangeable?):
The $T$ is indeed transpose. The reason it used is to define a column vector using less space.
The equality for $y$ is a dot product between two vectors defined as $$u\cdot v =u^T v$$ which coincides with a matrix vector product (the matrix is $n\times 1$ in this case and a vector $1\times n$). Particularly you have $$u\cdot v =u^T v= \sum_i u_i v_i =\sum_i v_i u_i = v^T u =v\cdot u$$