That's for you to decide in defining the notation you use, or to clarify by using parentheses.
I personally would tend to take $u\cdot\nabla u$ to mean $(u\cdot\nabla)u$, because I studied physics and that construction is more common there than the other one. Also, a dot is much more usual for the scalar product of two vectors (the first interpretation) than for the product of a vector and a matrix (the second interpretation). But still it's somewhat ambiguous, and if you can't expect your readers to know which one you mean, parentheses seem advisable.
Anyway, $\nabla u$ is a bad (though common) notation for the gradient of a vector field, since it doesn't fit with the usual rules for matrices and vectors. The “right” way to write it would be $(\nabla u^\top)^\top$, and then there would be no confusion, since $(u\cdot\nabla)u\neq u\cdot(\nabla u^\top)^\top$ and the only consistent way to interptet $u\cdot\nabla u$ would be $(u\cdot\nabla)u$.