Pleasing shapes have been found in every civilization throughout history. Many of them share the same basic rules of good design that were first taught in ancient Greece, and used for 3000 years of pottery and furniture design. We can also apply these rules to the things we turn from wood.

The "Rule Of The Golden Mean".
The Greeks perfected and used the "Golden Mean," a formula for the ratio between the short side and the long side of a rectangle that will appear balanced to the viewer. I learned to use the "rule" for furniture design back when wood shop (Manual Arts) was still a required subject in high school.

The Rule Of The Golden Mean simply states that:

"The smaller is to the larger as the larger is to the whole."

Without a lot of discussion, this can be reduced to: "The relationship between the smaller and the larger is the ratio of 1 to 1.618". To complete the "rule" the sum of the length and width is the same as the length multiplied by 1.618.

We can find examples of the rule in the sizes of tables and other furniture. A 72" long dining table is 45" wide. An oval coffee table that is 42" long is 26" wide. The 60" round dining table still has a balanced proportion with a 36" leaf inserted in its center. A bookcase that is 72" tall will be 45" wide. All of these examples fit the "Golden Mean", the larger (length) is the smaller (width) multiplied by 1.618.

To apply the "Golden Rule" to a bowl design, we divide the diameter by 1.618 to determine its height and divide its height by 1.618 to determine the diameter of its base.

As an example: If we want to turn a 10" diameter bowl, its height will be: 10" ÷ 1.618 = 6.180", or 6¼" height And the base diameter is: 6¼" ¸ 1.618 = 3.72", or 3¾" These dimensions will give us a bowl of pleasing proportions, as taught by the ancient Greeks. What we do with the curves between these dimensions is the topic for another discussion.
Design Rules
Rules-Of-Thirds
There are other rules of proportion that we can use. One of these is the "One Third-Two Thirds" Rule that is easier to use and similar to the "Golden Mean" within the dimensions that we would use for a bowl design.  I use this one because most things are easy to divide or multiply by 3.
Rule 1 -
"The bottom diameter is 1/3 that of the largest diameter."

Rule 2 -
"The height is either 2/3 (preferred) or 1/3 (optional) that of the largest diameter."

Rule 3 - 
When the largest diameter is not at the top rim, it shall be located at 2/3, or 1/3 (optional) of the Height above the base. "
Most turned shapes have a more pleasing appearance when the maximum diameter is not at the top rim, but is located below the rim. This "Rule" tells us how far.

Rule 4 -
"The diameter of the rim, or top opening, shall be some multiple of 1/3 or 2/3 that of the Diameter."

Rule-4 (Optional for Bowls only)
"The diameter at the rim is smaller than the diameter by approximately 1/2 the distance that the largest diameter is below the top of the bowl."
I added this rule as a follow-up to Rule-3 after being asked for some guidance on how much smaller the rim diameter should be. I measured several bowls that looked pretty good to find something that they shared, and came up with this rule. It seems to work most of the time.
Russ' Rules
There are several other "rules" that I have learned, either from experience or from other woodturners.
Vases and Urns
The same Rules can be applied to these common forms as shopwn in the photo below. They always seem to look better when the "rules" are applied.
Concentrate on form, not wood grain.
A lot of wood turns darker with age. In the end, all that you may have left is the shape.

All curves must be "fair".
A fair curve is one that flows with smooth transitions as it changes in shape or from a concave to a convex curve.

There are no flat areas between curves.

Lift the object off of the table or other surface on which it is sitting.
Ideally, it should appear to be floating slightly above, rather than being firmly attached to the surface. With the bowl or vase sitting on a table, follow an imaginary continuation of the lines of the sides of the bowl or vase until they have intersected under the bottom of the vessel. This intersection will be inside of the foot of the piece. It should be above the table surface for the vessel to appear as being lifted above it.

There is a movement among woodturners to not have a foot on the bottom of a bowl. Making it appear to be lifted from or setting lightly on, the table is most difficult when there is no foot to work with.

Inverted appearance should be as pleasing.
Turn the bowl or other turned object upside down on the table. Its proportions should be just as pleasing to the eye as they were in the upright position. If not, there is something wrong with the proportions or the curves between them.

The Outside Space is just as important
Set the bowl or other turned object against but a short distance away from a vertical surface or line, and imaging the space between the bowl and the wall having a mirroe immage. Concentrate on that shape. Take a photo if necessary. The proportions of the space will be the reverse of the outside of the bowl, but it should have the same pleasing appearance. If it doesn't, there is something wrong with the bowl.

Look at the space between pairs
This is similar to the previous "Rule" except the space is defined by an identical pair of objects.

When any pair of objects, vases, candle holders, etc., will be viewed at the same time, the shape of the space between them is as important as that of the objects themselves. Always work their design as a pair, never as individual pieces that are later placed together, and always consider the shape of the space between them.

Recognize that indentical objects, such as candle holders, can be separated at a considerable distance and the shape of the space between them will still influence how we view  them.  I would venture to say that, as long as we can see both objects within the frame of our vision, the space between them will be important as long as there is no other object between them.

When there is an object separating the pair, then the space between each of the pairs and the object has to be considered separately. An example of this would be placing a flower arrangement between a pait of candleholders. 
Application of the "Rules"
We can apply these rules to our bowls, vases, goblets, and whatever else we might turn from a piece of wood. I am using the "Rule of Thirds" for these illustrations because it is easier to divide any dimension by 3, and it works for the shapes and sizes that we turn on our lathes.

I don't always measure whatever is being turned, but I am always making a visual comparison of the object to the "Rules" because I know that I will like the results, and that others will also like them. Sometimes I break all of the "rules" but I know where I started - "The Rule of Thirds".

Bowls
The Rules are easy to use for a bowl. All of the rules were followed in the bowls in the following photos. 
Goblets
Goblets are an excellent example of the "Rule of 1/3's". Everything is related to everything else with a relationship of 1/3 or 2/3. Using the photo of a goblet, the overall height has been divided into 2 parts. The globe of the
goblet A-B is 1/3 of the overal height A-C, and the length of the stem is 2/3 of the height A-C. The globe and the stem are then farther broken down asshown. The diameter at D is 1/3 the height A-C, and the diameter of the base at C is the same as the diameter at D. The bowl (A-B) and the stem (B-C) are thurther divided into 1/2 and 2/3 as shown.  E is the location of the smallest diameter of the stem.
The relationship of the bowl and stem are more defined with a more conventional "wineglass" shape.  On a goblet like that shown here, it is not as defined, but we whould still try to keep these same parameters for a more pleasing shape.
The 1/3 and 2/3 relationships of the stem and globe have been reversed.

The relationships of the diameters and their locations within the stem and bowl are also reversed.

The diameter at (D) is still 1/3 that of the overall height (A-C), and the base (C) is still the same diameter as the widest part of the bowl at (D).
Again, a note on the base diameter at (C).
This has also been reversed. The shape of the short-stemmed goblet has the opposite effect on the appearance of the base (C). Here the shape of goblet makes the diameter of the base at (C) appear to be smaller than it really is. This has the effect of making the goblet appear to be top-heavy. To prevent this appearance, it is important the diameter at (C) be no smaller than that at (D); and it wouldn't detract from the appearance of the goblet if it were slightly larger. Again, how much larger is a judgement call. The base of the goblet in the photo is about 3/16" larger than the bowl at (D). A little bit has a large effect on the overall appearance.
A note on the base diameter at (C).
The relationship between the bowl and the stem will influence the appearance of the base, and therefore the overall "balance" of the goblet.  The long-stemmed goblet in Figure-xx has the effect of making the base appear to be a larger diameter than than it is.  This illusion can be corrected by making the base diameter (C) slightly smaller than (D). How much smaller is a judgement call that depends on the style of the goblet. The base is about 1/8" smaller than that of the bowl (D) on the goblet shown in the photo.
Everything about this bowl is related to everything else by ratios of 1/3 or 2/3

Diameter of the bowl is 12"

Rule-1, Base diameter is 1/3 Diameter (4")

Rule-2. Height is 1/3 Diameter (4")

Rule-3, Diameter is located at 2/3 the Height of the bowl (2-5/8")

Rule-4, Rim Diameter is less than the Diameter by 1/2 of 1/3 the Height
  (11-3/8"), and I will let you do the math on that one.

Bowls, Alternate Height per Rule-2
The Reverse Option of Rule-3

This an example of what happens when the Rule No. 3 has been inverted to locate the Diameter at 1/3 of the Height above the Base.

The Height does not follow Rule-2, and it is approximately EQUAL to the Diameter.
Another Inversion of the Rules
The "rules" can be inverted for the Gobl;et, just as we did with the bowl forms. Here, all of the proportions are reversed. Most people like this better than the more normal form shown in the above photo.
Other Forms
We can apply the Rules of Thirds to almost anything we can turn, from Candlesticks to Weed-Pots. Here is how they are used on a simple candleholder.
Again, everything is related to everything else by a ratio of 1/3 or 2/3.

Sometimes it is necessary to make the Base Diameter smaller than 1/3 the overall height because the larger wood might not be available. The smaller Base looks just as good if the same proportions are used  to determine the diameters at B and C.
----------------- Base
--------------- Rim
---------- Diameter

Everything about this 12" bowl is the same, encept that the optional Rule-2 was used, and the Rim diameter (Rule-4) was changed to match.

Diameter is 12"

Rule-1, Base is 1/3 the Diameter (4")

Rule-2 (Optional), The Height of the Bowl is 2/3 the Diameter  (8")

Rule-3, Diameter is located at 2/3 the Height (5-1/4")

Rule-4, Rim Diameter is smaller than the Diameter by 1/2 of 1/3 the Height
and I will let you do the math on that one  (10-3/4")
--------------------------Rim
-----------------------Base
Rule-1, Base diameter is 1/3 the Diameter

Rule-2, Height is 2/3 the Diameter (Optional)

Rule-3, Diameter is located at 2/3 Height above the Base

Rule-4, Rim diameter is 1/3 the Diameter. This does not fit the Rule as
--------------Diameter
2/3 Height
1/3 Height