For more (much more) general information, I recommend the Nikon site and the Olympus site.
The above photos show a very small part of the Nikon objective offerings that have been available over the years.
While each component in the system is important to image quality, the single most critical optical element is the objective lens. As seen from the photos above, there is an incredible variety of objectives, optimized for various tasks. A list of codes used by various manufacturers is given on Nikon's Microscopy U. There is also a dedicated page about specialized objectives there.
There are essentially seven critical parameters for any objective, as well as other performance parameters, such as chromatic aberration and field curvature correction, as well as specialized types, such as phase contrast, modulation contrast, and interference. The seven to keep in mind determine which microscope system and what type of specimen can be observed.
In general, the older objectives are for finite tube lengths, such as 160 or 170 mm (diascopic) or 210 mm (episcopic), whereas modern higher-end systems primarily use infinity corrected objectives.
Nikon objectives from the 60's and 70's are often of the "S" type, for use with the Model S system, which is characterized by short barrels with short parfocal distances. They are mostly not well corrected for color or field, but there are exceptions. The designation "M" means metallurgical, which is used to look AT objects, rather than through them (episcopic illumination), so typically have tube length requirements of 210 mm. They have RMS threads and a parfocal distance of 45 mm.
Simple objectives may have no designation other than the magnification and NA. I refer to these as "ROMO"s, (Regular Old Microscope Objectives, of course!)
Certain manufacturers, in this case, Amscope, make RMS threaded infinity corrected objectives without fancy corrections.
Modern Nikon infinity corrected objectives have an M25 thread. The standard set is an "E Plan".
One of the metrology/inspection manufacturers, Mitutoyo, makes some spectacular infinity corrected objectives featuring very long working distances. They are quite large and have an odd thread (M26, but 0.706 mm pitch instead of the BD pitch of 0.75 mm), so are not easily used. However, they make incredible "extreme-macro" lenses. You can see the size of them in comparison to a Model S, both at 10x power.
Returning to finite objectives, the next step up from ROMOs are the Plan objectives, which feature a flat field (the edges are in focus simultaneously with the center of the field).
While counter-intuitive, it is actually fairly difficult to have good low-power imaging. 1x and even 0.5x Macro Plan objectives exist, but the lowest I have is 2x, one of which is a Plan Apochromat, which is a gorgeous lens.
As mentioned, the "M" objectives are for looking at the surface of objects, so use episcopic illumination, in this case, requiring a tube length of 210 mm. Also, for bare surfaces, no cover slip is used, so they are referred to as 210/0 objectives. Below is a set of M Plan objectives ranging from 5x to 100x.
Another specialized objective are the BD Plan, or the similar ED Plan (extra-low dispersion), which have a segregated outer annulus for reflected darkfield illumination, where the incident light goes through the outer barrel and the imaging light goes through the lower NA inner barrel. These are what I use for reflected Rheinberg as well as truly spectacular darkfield.
Phase contrast requires a specialized objective as well, with a annular phase ring in combination with neutral density, where the density determines the contrast. Dark low low, dark low, and dark medium, create an increasingly dark background, for example. A new type, the ADL, which is apodized dark low, seems to only be available with an infinity corrected objective, reduces the characteristic halo effects which are undesirable.
Hoffman modulation contrast also requires an objective with something inside to match the condenser.
One of the methods for excellent color correction is to use special glass types, including fluorite correction (fluor). These typically also feature extremely high NA for the objective power, making it ideal for epi-fluorescence, where you want to capture as much of the light as possible. Plan Apo lenses also tend to have high NA (the 60x PlanApo shown is a truly spectacular lens, but requires oil immersion to get the full NA of 1.40.) They can also be made as phase objectives along with the fluorite correction, with great results. The objective which is the second to the left shown below is a 10x fluor Ph2 with an NA of 0.5, which is double the NA of typical 10x objectives. The leftmost one is identical, but without a phase ring.
To do interference microscopy, of course requires special objectives, too. There are several types, these shown are 10x and 20x DI, which are Mirau objectives, and a 40x MI, which is multi-beam interference.
The final one I currently have a photo of is borrowed from a friend. It is a strange looking beast, designed for water dipping, allowing penetration to the bottom of a water well up to about 1 cm deep. It has an inherent NA of 0.165, but multiplying that by the index of refraction for water of 1.33, gives an NA of 0.22.