Question
We have a problem with standing water in back yard, next to where our patio (made w/pavers) was built, the water is collecting on far right side of yard and leaving a large portion of standing water and muddy ground, so when we get grass cut, it looks great on one side and like a muddy mess on the other side. . . . We have tried using those black tubes to take addn'l water from roof away from house and our neighbor who lives next door has been re-directing their water from running off that side of our yard to keep excess water from flowing onto our property, but water (especially during heavy rain still continues to collect and stand their looking terrible).  I don't even get back yard grass cut now (since it looks so bad when the lawn mower runs over that muddy area and leave long muddy lines of grass and mud on that side of the yard). . .

Answer
Sounds simple. You need to have your yard regraded. Call some one with a bobcat and some fill dirt. By default (and physics) If water is collecting then you must have a small depression or a low spot that needs to be raised and sloped to drain the water off.
You could also try installing a french drain through the wet area.
Search "french drain" on the Internet to learn how to do that.