Do these matrices have any name? Assume $A$ is a square matrix defined as follow: $$A=\sum_{i} u_{i}u_{i}^T$$ where for each $i$, $u_i$ is a non-negative column vector.
Do the matrices of these forms have any special name?
 A: Yes, these are the completely positive matrices.
A: There is a problem with the term 'non-negative column vector'. The notion of non-negative is defined for square matrices.
We can require $u_i$'s such that $M=[u_{ij}]$ is positive (non-negative) that is such that $M=UU^*$ for some square matrix $U$.
There is also a different notion of positivity (again for square matrices) which requires the entries to be non-negative.
One vaguely related idea is the following. If $M$ is a symmetric matrix that is orthogonally diagonalised by 
$P=[u_1\space{} u_2\space{}\dots u_n]$
and if $\lambda_i$'s are eigenvectors of $M$ corresponding to the unit eigenvectors $u_i$'s, then
$M=\lambda_1 u_1u_1^T+\dots +\lambda_nu_nu_n^T$ 
is called the spectral decomposition of $M$.
A: Let's assumes there are $n$ such $u_i$'s and there are exactly $k$ linearly independent vectors among them, then matrix $A$ would be a rank-k matrix.
Each of the $u_iu_i^t$ is a rank-1 matrix and they would sum up to a rank-k matrix if exactly k out of them are linearly independent.
 As $x^t(u_iu_i^t)x=(x^tu_i)(x^tu_i)^t \gt0$ , each of the $u_iu_i^t$ is a symmetric positive definite matrix, thus A is also a symmetric positive definite matrix.
