Perhaps the question is self-explanatory. The context is Kleiner's Inv. Math. paper An isoperimetric comparison theorem, where the statement of the main theorem begins with "Let $M$ be a complete one-connected three-dimensional Riemannian manifold...".
The article does not define what one-connectedness is, and google is not much help either. The article makes reference to books by Gromov(-Lafontaine-Pansu) and Burago-Zalgaller, and a JDG article by Aubin, where maybe this is defined, but none of these references are available to me now.
(I know that we say that a subspace $A\subset X$ is $k$-connected if the relative homotopy groups $\pi_\ell(X,A)$ vanish for $\ell\leq k$, but here there doesn't seem to be a distinguished subspace.)