If you're talking about where to measure when reading size charts, I believe the bust measurement is meant to be your band size, waist is meant to be smallest part, and the hip measurement is meant to be the widest part of your hip, wherever that may be, and it would include part of your bottom.
This is why jeans sizing never makes any sense. I take a 25 jean even if it is low rise and obviously the waistband would be larger than 25 inches. And of course size charts don't take into account vertical proportions, except in the most rudimentary of ways.
If this is of interst to you, you can google around, there's a lot of interesting blogs talking about the "science" of size charts for women, and how they were developed, and why it's such a challenge to come up with standardized sizing. There is one blog that goes into amazing depth; the author analyzed old patterns from back in the 1800s to present, I can find the link again if you are interested.
If you're talking about where to measure for tracking fitness and health, then it helps to measure multiple places including:
1) under armpits
2) bra band
3) over the bust (i.e, cup size)
4) at smallest part of waist (where you bend)
5) at the belly button, while sucking in and pulling the tape measure as tightly as possible: this is a measurement of internal fat, the dangerous stuff
6) at high hip - top of your hip bone
7) at widest part of hip
8) at widest part of thigh
9) calves
10) biceps
11) Neck
#5 and #6 are what are used to calculate one's waist-to-hip ratio, an indicator of whether you are an apple or a pear. A lot of people think you use #4 and #6, and that's incorrect.
#9 and #10 can really impact your calculation of body fat percentage, if you use those online calculators. You don't necessarily want this number to get smaller, because the calves are the strongest muscle in the body. If they shrink too much, you may be losing muscle, not fat.... Of course this is a general statement, because most people can tell if these areas are larger due to fat vs muscle! Just saying, if your calves are large because they are muscular, don't hate them. This is a GOOD thing.
And #11 is used by the Army/Navy body fat calculators. My neck measurements never changed, but I certainly grow more than one chin when I gain weight.
Finally, wrist measurement can be used to determine frame size. This is because you don't usually have any fat around your wrist, so you are measuring your bone structure instead.
More detail than you probably wanted to know... yes, I've spent too much time thinking about this myself.