In the most general terms (I'm not asking for the actual calculations), how do popular planetarium apps and software calculate the positions of celestial objects? Planets, for example. Do they use Keplerian formulas with their associated elements and rates, or do they calculate long series of periodic terms as outlined in chapter 32 “Positions of the planets” in Meeus? I'm assuming positions of comets and other small bodies would be calculated from Keplerian orbital elements? I'm also assuming they don't use some sort of numerical integration to calculate positions (as do JPL Horizons)?
There are quite a few different types of methods of computing the position of celestial objects, and the method used to compute the position generally depends on the type of object and how accurate the application needs to be. I'll list a few types and methods below.
Nebula These are large, distant objects. They generally don't move fast, and are large enough that highly accurate positions are unnecessary and/or difficult to define. So these are generally just static positions from a catalog like the NGC or Messier catalogs.
Stars Distant stars are pinpoints of light, and easy to agree on a location and measure movement, so a star catalog generally includes the position of the star at some epoch, and includes proper motion fields to show how the star moves over time. The exact data is catalog specific. For example, the HIPPARCOS catalog contains the star positions for epoch 1991.25, and proper motion fields specifying their change in RA/Dec from that time.
Planets Lots of methods for computing the planets' positions have been developed because of their importance and popularity.
Keplerian elements are one way, but become inaccurate rather quickly. But the Explanatory Supplement to the Astronomical Almanac supplies elements, and some extra terms to help with accuracy over time. An implementation is here.
VSOP87 is probably the most popular method as it provides good accuracy over a long period of time and doesn't require much storage. Implementations are available in a wide array of languages. This is likely the ephemeris used by pretty much all applications you can download to your PC or phone.
JPL Development Ephemeris are highly accurate, in fact the most accurate available to the public. They are generally produce by the JPL as needed for specific missions. They use Chebyshev polynomials to represent the positions which are only accurate for a very short period (the longest is 32 days), so to cover long time ranges, a lot of these need to be stored. E.g. DE422 covering 3000BC to 3000AD is 500Mb in binary form. Due to their size, and lack of need for such high accuracy from a planetarium program, they are usually only used for specialized applications. I have written an article Format of the JPL Ephemeris explaining how to use them to implement your own, and provided implementations in a few languages.
Artificial Satellites Many planetarium programs also include artificial satellites. These are based on Keplerian elements, but quickly go out of date, so a more complex model is used to adapt the positions called SGP4/SDP4 (and SGP8/SDP8). The algorithm was made public as source code from NORAD in the 80's, and code is available in a multitude of programming languages, but the most thourough ones are available at Celestrak. The elements also need to be updated regularly, and Celestrak provides some of those, but Space-Track is the official source. Programs have to update this file on at least a daily basis to stay up to date.
Comets/asteroids Minor solar system bodies are of enough interest that they are tracked, but not important enough to generate the effort the JPL DE requires. Instead, new Keplerian elements are generated and distributed by the Minor Planet Center. Again, these have to be updated quite regularly, but not necessarily daily, so most apps leave it up to the user to request an update.
Moons The Earth's moon is included in the VSOP87 and JPL DE. And the moons of most planets have chaotic enough orbits that their importance/difficulty tradeoff doesn't justify a specialized ephemeris. But you will find algorithms for some of Jupiter's and Saturn's moons in Meeus' book.
Earth Orientation Parameters are something you didn't ask about, but are quite important to determine where an object will appear in relation to an observer on Earth. Meeus' book covers some of these such as precession, and nutation, and other effects like aberration. But the Explanatory Supplement to the Astronomical Almanac provides a more complete and updated explanation. The USNO provides an example implementation as NOVAS.
How do planetarium apps and software calculate positions?
The generic answer is that they use some form of ephemeris. The VSOP87 is an example of an ephemeris. So is JPL Horizons, which uses ephemerides rather than numerical integration. While the ephemerides used by JPL Horizons are calculated using numerical integration, once the data are placed into the form of an ephemeris, numerical integration is no longer needed. JPL Horizons uses ephemerides calculated by other groups at JPL. That they're using ephemerides is why JPL Horizons is so fast.