AF with manual override means that you can use the AF to get a lock onto a subject and then you can manually make adjustments to that focus to perfect it. It generally means working with focus and recompose - where you use an AF point to focus on the subject (typically the middle point) and then recompose the photo for the final composition; which likely shifts the plane of focus, thus needing the manual override to make a slight adjustment, without having to hunt for the AF/MF switch on the lens itself (which can be a pain if your lens has more than one switch).
As said above though, experiment with the outer AF points as well (typically you want AF set to a single point, rather than all points being active) though the speed of them will vary, on entry level DSLRs they are weaker, whilst the middle point is the strongest - on midrange and upper level bodies the number of AF points increases and the outer points are also stronger/faster AF sensors as well.
It also works well if you combine it with backbutton AF control; whereby autofocus is not engaged by a halfpress of the shutter button, but by a press of a button on the back of the DSLR. This mean that you can use an AF mode which supports continuous AF (ie all the time you hold the button down the camera constantly checks and focuses, good for moving subjects) and if the AF is failing you just lift the finger from the backbutton and shift to manual focus instantly (no hunting for the AF/MF switch) and when you press the shutter fully the AF won't kick in and try to refocus again (thus letting you take the photo).
Combined with AF and MF the manual focus override/fulltime manual focus lenses are a very powerful tool and certainly capable of a lot. Learning AF will be important as the modern DSLR is made with the intent of AF being used; rather than the older manual focus cameras where manual focus aids and a brighter viewfinder were the norm since it was all manually driven.
As an additional point regarding AF remember that AF works by finding the closest point it can to the camera (under the AF point(s) which are active) where it can detect a contrast difference in the scene. Thus if you try testing on a pure white wall/blue sky or any other scene where there is no or very little contrast change then the AF can fail; another good example is foreground elements, eg if you are ever at the zoo and try taking a photo of the big cats, where there is typically a wire cage at a distance between you and the cats the AF can end up locking onto the wire rather than the cat (even though fully focused on the cat the wire blurrs out).
(ps big cats are a good example as often there are 2 barriers so that you can't put the lens right up to the wire to eliminate this problem).