Type was first knowingly modernized in the early 19th century.
It was understood at the time that technical improvements in printing necessitated a new style of type: it was described as modern at the time (more accurately now "didone"), although not named as such. Foundries then named their basic text types solely by the size.

As well as the new didone style there were other changes, most notably dropping the long "s" and introducing lining figures.

However, all the old type was melted down, so it wasn't around to be called "old style".
That comparison came in the mid 19th century, with the introduction of the modernized old-style, which was not a strict revival of any particular historical type, but a version of the modern type with certain old-style features -- including old-style figures, but not necessarily. It was called old-style, while the "basic" text type continued to be named by its size, or as just "roman": the modern was the default.

I don't know when old-style figures were first referred to as such.

Interestingly, when Futura was first published in the 1930s, it had old-style figures.

Now, with the InDesign OpenType palette suggesting that all typefaces should have both old-style and lining figures, even if that creates anachronisms, the picture gets a bit post-modern. For instance, I have included old-style figures in Scotch Modern, which is a facsimile-class revival of a face that would never have had them.

I prefer the term text figures, since their temporary eclipse by lining figures in the nineteenth century doesn't make them old per se, just out of fashion for a while before people rediscovered that using lining figures within text is the equivalent of all-caps within text, i.e., shouting. I prefer to use lining figures in titles and other all-caps settings, numeric tables, folios, and other places where they stand apart from the text proper, and to use text figures otherwise.

Effectively, lining figures are upper-case figures and text figures are lower-case figures, names I've not heard used much for them but that better describe their purposes.

As long as we're distinguishing cases of figures, a far rarer case of figure that I use in a special circumstance is the small-caps figure - numerals scaled to work in strings of small caps. This setting was pointed out by Geoffrey Dowding in Finer Points on the Spacing and Arrangement of Type, namely when using strings entirely composed of small caps as display matter, such as in ads or in secondary or tertiary subheads. In these settings, text figures don't blend with the small caps much better than lining figures do, since none of the rest of the line has ascenders or descenders. However, although I do use small-caps figures with small-caps letters in these special situations, normally I pair small-caps letters with text figures. When small-caps letters are used in their usual setting, i.e. to represent acronyms or other secondary forms of emphasis within text matter, it's better to pair them with text figures, which blend better with the lower-case letters that make up the bulk of the line. The recent Linotype faces (such as Optima nova and Palatino nova), Lapture, Tabac, Underground, and Mrs. Eaves include small-caps figures; I wish more would.

I'm reminded of the good discussion of small-caps figures here: http://typophile.com/node/29580

Miss Tiffany pointed out that Dolly includes them.

That discussion also includes some interesting critique of mixing small caps with text figures.

And now I see I've pointlessly hijacked the thread building a list of typefaces that already exists elsewhere: http://typophile.com/node/70865

I now return you to your regularly scheduled discussion of text figures. :)

I tend to call them "lowercase figures", since you are most likely using them with lowercase or u&lc text settings.

There was an Adobe publication (broadside, Font & Function, was it?) a few years back where it was mentioned that when signage became an important part of printing in the latter part of the 19th century, a lot of the advertisements used full caps. SPECIAL TODAY! etc. The numbers of the time -- "old style" figures = looked bad with those full caps. So "Modern" figures became standard. From a timing perspective, "modern" figures are just as modern as the hoop skirt.

As with so much for too long, advertising rules the type world.

I have a copy of an old version of "Teach Yourself Calculus", in which there are a few pages with log and trig tables... set in old style figures. In the metal type era, at least, digits were monospaced whether they were old style or lining, because many typefaces came with only one kind of digit (although there were also several where both were available as alternates) and you could never be sure that at some point tabular matter would not be needed.

With today's technology, providing additional alternates is much easier, and so getting away from monospacing, at least with oldstyle figures (also doing so with lining figures, even lining figures with small cap height, sounds very bold and adventurous to me) is now an option open to consideration.

But given the precedent that digits were monospaced just about universally (except in the more outré display typefaces) a monospaced version should always be available as an alternate, I would think.

@charles_e: Yes, there were a circa 1989-90 issue of Font&Function with a good article on the historical adoption of the different sets of numerals.
Adobe contributio (also with their types) has been pretty important in re-introducing the sensibility towards an appropriate use of the two forms.

Adobe Garamond (1989) was an important typeface, making all the bells and whistles of traditional typography available in the digital realm.

However, FF Scala (1990) was the first digital typeface (and a new design, at that) to have old style figures as the default.
The lining figures were in its Capitals font.

Wasn’t also FF Quadrat? I have to check, but I am pretty sure I never had to switch fonts (I have the old Type1 license).

FF Meta also had OSF as default.
And Emigre's Matrix (very popular in the early '90s).