These dualities stem from the derivation of type from written forms, and represent two historical ways in which these letters developed prior to the invention of printing from moveable type. These are not, by the way, the only forms of g and a encountered in palaeography, there are other forms which did not influence the development of typographic forms, or whose influence was short-lived or localised. A good example of localised influence is the traditional Irish g. You can see some examples in Michael Everson's digitisations of historical Irish types.

I think John did a good job of answering the question concisely. But I'd like to add a little more, if I may

Correct me if I'm wrong, but...

The double-storied g and a are commonly features of serifed roman fonts, wheras the single-story forms are more characteristic of an italic or a cursive font (or writing style).

for your consideration: does the following glyph look more like the double-storied g? or a single-story g?

I think the distinction is really about how the bottom part of the g connects to the top part, and what we think of as single-storey is a class of form in which the bottom part of the letter comes off a right stem.

here's another interesting question related to these 2 letters... take a look at Paul's post (#8) in this thread: http://www.typophile.com/forums/messages/4100/68949.html?1113004951.

what do you make of this? is it really harder for young children to "read" these letters? or is it just that we teach the simpler forms first?

Nobody has addressed printing problems of the various g's which have had so much influence and about which there must be many who can speak much more authoritatively than me. The major reason for the simplified g of Helvetica (and the a) is that they are much less likely to clot, not just in metal, but also in digital type, at small sizes. (Look at the Meta bold g in Fowler MEU 3 for a terrifying example of this problem in modern printing and this using a supposedly rational font. Why does anybody ever believe anything that a manufacturer or designer ever says about a font?)

One miscellaneous point: many metal fonts with beautiful g's have had short lives and limited usage for the simple reason that the character broke too easily.

is it really harder for young children to "read" these letters? or is it just that we teach the simpler forms first?

I was talking about this with Sue Walker, head of the typography department at the University of Reading and an expert on the typography of childrens' books. She says that there is no evidence that children cannot easily recognise and accept two forms of a/g, even at an early age.

Hi Paul,

"Correct me if I'm wrong, but...
The double-storied g and a are commonly features of serifed roman fonts, wheras the single-story forms are more characteristic of an italic or a cursive font (or writing style)."

This is generally true, though there are certainly always exceptions. Double-story "a" and "g" are also fairly common in sans serif Roman fonts as well. But of course, in an effort to simplify the structure, many sans serif designers opt for a single-story character.

"for your consideration: does the following glyph look more like the double-storied g? or a single-story g?"

As always, this is one of those cases where one could argue it falls under either category. (Sort of like, is the movie a drama with comedy, or a comedy with drama.) Within the branch of Gothic characters, it was also quite common to find "a"s that appear to be in both categories at the same time.

But if one wanted to be a historic stickler, the"g" example you show would more technically come under the heading of a single-story character. This is because it is a Textura character following the same principle as the Italic (and other manuscript hands of that era) lowercase "g." The lowercase "g" of that time was always closed (i.e., the descender hooked back up to the nadir of the body). This was done to prevent it from being confused with a "q," which back then was written like we make today's "hooked-g." This enclosed Italic "hooked-g" was a single-story "g" (even though it appears to fall into either category) because within the same Italic alphabet there was also a version (sometimes multiple versions) of "linked-g" (double-story g) as well. Both "g"s were freely used side by side.

As always, this is one of those cases where one could argue it falls under either category.

exactly what i was getting at. i think especially looking at the scripts that predated printing types you can see where either of our modern forms of the letter could be derived from the same script form.

But of course, in an effort to simplify the structure, many sans serif designers opt for a single-story character.

what i think is really interesting is that the sanserif italic has particular freedom in adapting any of the letter styles for it's particular needs. The first time I saw it, I was fascinated by Bringhursts table of the Italicization Quotient of Thirteen Unserifed 'Italics' in Elements.