Spectral Types and Classes
While there doesn't seem to be an "official" distinction between spectral types and spectral classes, I'm making a distinction in this blog; to wit, the spectral type is the alphabetic designation (e. g. O, B, A, F, G, K, M, and certain others), whereas the spectral class is the spectral type with the numerical value appended (e. g. G2, F9.5, etc.). We might think of these in the same terms as biologic taxonomy, with the spectral type corresponding to the genus and the spectral class corresponding to the species.
In this system, stars are organized according to their surface temperatures using what is commonly known as the Harvard System, with seven distinctive groupings: O, B, A, F, G, K, and M (often remembered by students with the mnemonic (Oh, Be A Fine Girl/Guy, Kiss Me). These were first codified in the modern order by Annie Jump Cannon between 1901 and 1912, and adopted formally by the International Astronomical Union on May 9, 1922.
This is a one-dimensional system, which concerns itself only with the characteristics of the spectra of the stars. It has been expanded to include spectral types L, T, and Y, which classify the infrared spectra of very cool stars (to include brown dwarfs), as well as types C for carbon stars (red giants near the end of their lives) and S for stars falling between M and C and which have excess amounts of zirconium. 
The Morgan–Keenan (MK) classification system was developed by William Morgan and Philip Keenan, and expands the Harvard classification, adding a luminosity class (see below), notated in Roman numerals, from 0 to VII. This two-dimensional classification scheme relates to spectral lines and surface gravity (which is related to temperature), whilst the Harvard classification is based on just surface temperature. Differences in the spectra of stars can thus be interpreted as luminosity effects, allowing analysis of the spectrum to lead to the assignment of luminosity classes.
The H-R Diagram is a scatter plot which maps surface temperature against brightness, and is thus a two-dimensional system which synthesizes the Harvard system and the MK system. It was created around 1910 by Ejnar Hertzsprung and Henry Norris Russell when they were exploring the relationships between different characteristics of stars. The H-R Diagram was a major advance in helping astronomers to understand the processes of stellar evolution, and thus helped pave the way for our modern understanding that stars are not permanent, but have a lifetime during which their characteristics change.
Below is a table which lists the specific temperature ranges associated with the various spectral types:
Sc = the numerical portion of the spectral type designation;
M = a constant value representing the maximum temperature for the spectral type in question (in Kelvin);
T = the actual surface temperature of the star in question (in Kelvin);
K = a constant value associated with each spectral type
The M and K values for each spectral type are given in the following table:
For example, the Sun's surface temperature is listed in most sources as about 5800 K. (The actual figure is closer to 5839 K, which we'll see shortly.)
However, if we didn't already know that it's a G-class star, we could consult the first table, and find that 5800 is in the G-class range. We can then use this value in the equation for spectral type G:
Conversely, if you already know the numerical portion of the spectral class, you can calculate the exact surface temperature of the star in question with the following equation.
Thus: the Sun is typically listed as a G2 spectral class; searching the table for spectral type G, we arrive at M = 6000, and K = 1.249 ⨉ 10⁻², and so the precise value for the surface temperature of the Sun (or, indeed, any star with a spectral class of precisely G2) is:
As a further example, the star Zeta Tucanae is listed as F9.5.
Consulting the first table, we see that spectral type F has a temperature range of 6000—7500 K. Using the numbers from Zeta Tunanae's spectral class in the appropriate equation from the table above:
A look at the H-R Diagram above reveals an important fact--stars of the same surface temperature are not necessarily of the same mass or the same radius. Pollux (β Geminorum), for instance, is in the same temperature range as Epsilon Eridani (ε Eridani), but Pollux is a giant star in the vicinity of 10 times the radius of the Sun, whereas Epsilon Eridani is a Main Sequence star slightly smaller in radius than the Sun. They are both of spectral type K, and their spectral classes are similar—Pollux is K0 while Epsilon Eridani is K2—but this cannot be taken to mean they are of similar mass and size.
This problem is handled by luminosity classes.
Luminosity classes were established to handle the problem of stars of differing sizes and masses having similar spectral classes. A star may be of the same temperature as another star, but if the first one is significantly larger, it will be more luminous, simply because it is emitting its radiation from a significantly larger surface area.
So, while luminosity classes do not directly refer to a star's size, they point indirectly toward it.
Luminosity classes are designated by Roman numerals (sometimes with an additional lower-case letter following), and are appended to the end (with two exceptions) of the Spectral Classification, sometimes with a space between and sometimes run together.
Below is a list of the luminosity classes:
• 0 or Ia+ (hypergiants or extremely luminous supergiants)
• Ia (luminous supergiants)
• Iab (intermediate luminous supergiants)
• Ib (less luminous supergiants)
• II bright giants
• III normal giants
• IV subgiants
• V main-sequence stars (dwarfs)
• sd (prefix) subdwarfs
• D (prefix) white dwarfs
Note that the last two, "sd" and "D" are put before the spectral class designation.
Note, also, that Luminosity Class V stars are actually the most common kinds of stars in the universe (I talk about the frequency of particular spectral classes in the next blog). They are termed "dwarfs" because the radii of the Luminosity Classes IV and below tend to be much larger: UY Scuti, the largest star yet found, is of Luminosity Class Ia and is estimated to have a radius 1,708 times that of the Sun. To give a sense of this size, if it were to suddenly replace the Sun, its surface would be located just beyond the orbit of Jupiter.
So, Pollux, being significantly more luminous due to its size being in the range of 10 times the radius of the Sun, falls into Luminosity Class III, and its full designation is K0III; whereas, Epsilon Eridani, being less luminous by virtue of possessing only 0.735 times the radius of the Sun, falls into the Main Sequence—Luminosity Class V—and has a complete designation of K2V.
* Because elements heavier than helium are created by the nuclear fusion in the cores of stars, there were no elements other than hydrogen and helium in the early universe; thus, any stars which might still exist from that time—before about 400,000,000 years after the Big Bang—would have few-to-no metals in their structure.
Thus, an alternative way of thinking about stellar populations is to think in terms of generations:
Note that this nomenclature is in no way official, nor do I expect to single-handedly revise cosmology by suggesting it; I included it because for the purposes of worldbuilding, it might prove useful.
The Next Generation?
Probably not. While there is certainly still star-formation going on in the universe (even right here in the Milky Way), the new stars forming are still Population I stars, because the gasses and heavier elements that go into their formation are not changing in proportions over time.
While supermassive stars are still producing heavy elements, and supernovae explosions are still distributing those heavy elements into the cosmos, the overall amount of such heavy elements is not increasing, nor are heavier elements being produced than those produced in the past. Therefore, newly formed stars cannot have a basic composition in any way fundamentally different than their older siblings among the Population I stars.
Indeed, if the "heat death" models of the universe are correct, star formation will eventually cease, some time around when the universe reaches age 10¹⁴ (100 trillion) years (that's about 72,000 times longer than it has currently existed).
Supermassive stars will explode and collapse into black holes; somewhat less massive stars will produce quark stars (possibly) and neutron stars (certainly), and even the lowest-mass main sequence stars will end their lives first as white dwarfs and finally as cold black dwarfs.
Beyond that, things become highly speculative, but the theories are worth a read for worldbuilders who are interested.