I'll give 'er a go. There are others here who could give you a technically-superior explanation, but I'm pretty sure I can give you a quick synopsis.
BHIP% calculates the ratio of ground balls that end up as hits (generally, that would be singles). Without going too deeply into the theory, BHIP% says that a pitcher's basic skill set will result in X% of ground balls ending up as singles and that number will be different for each pitcher. There will be inevitable variations, but the theory is that pitchers will tend to regress towards their natural mean if nothing else changes. When analyzing BHIP% numbers, you can check variation within a pitcher's BHIP% calculation to see if there is an explanation for that pitcher's stats. If a pitcher has a particularly low WHIP as well as a BHIP% that is below his personal average, you can bet that his WHIP will probably be on the rise soon provided that no other factors (namely, infield defense) have changed dramatically.
BABIP is a little more broad. Instead of focusing on ground balls that end up as singles, it focuses on all balls hit into play. Home runs are excluded from this calculation as they are never technically in play. BABIP suggests that there is a natural mean for all pitchers when it comes to the number of balls that end up as hits after being placed in play. I don't have the long-term history of the calculation, but I do know that it has been right at .300 for quite a few years now. The theory is that few pitchers have the ability to vary wildly from this average for an extended duration (Smoltz
is a good example of a great pitcher that falls into this average). You can use this stat in the same way that you use BHIP% by checking to see if a pitcher's current stats make sense given his BABIP ratio.
The easiest way to keep the two separate in your head is to change the name of BHIP% to singles average.