I'm barely out of Kindergarten on this subject myself so this one is for the raw newbies. I going to try to frame this in the way I would have liked to see it presented 6 months ago. Since my microscope is an older 160 mm T.L. Zeiss, most of my information applies mainly to that microscope. However each of the other "big four" follow a similar format.
Objectives are the heart of the microscope and likely the most important component in delivering a good image to the eyepiece or camera sensor. Objectives are built up with differently shaped lenses and usually the more glass, the better the image. Let's look at why that is. Lenses have three fundamental problems:
1. Different wave lengths of light (different colors) focus at different points out from the lens with red light being the furthest away and violet being the closest. This happens because each color has a slightly different refractive index and thus comes out of the back of the lens at a different angle. Of course white light includes all the colors between red and violet so your depth of focus is spread out rather than at a point, which is what we want. This is called Chromatic aberration and is generally corrected for to some degree in even the cheapest lenses.
2. Light rays coming to focus from the outer edge of the lens do so at a different spot than those coming through the center. This is called Spherical aberration and can be corrected, but at a cost.
3. A flat image projected through a lens will result in a slightly curved image. This results in uneven focus across the face of the image. For example, the center of the field can be in focus while the outer edge is slightly blurry. This can be fixed, again at a cost.
The least expensive lenses are the Achromats or Achromatic objectives. These are corrected for two colors; red and blue and also corrected for spherical aberration for the color green.These are OK for viewing but not so good for photography. If you must take a picture, best results can be obtained using a green filter and viewing the prints in B&W.
Next up are Planachromats or just plain Plan objectives. These are corrected similarly to the achromats with additional correction to create a flat field. These are the minimum that should be considered for photography.
The Neofluars, sometimes called semiapochromatic, are another step up. These lenses have flourite elements and are corrected for 3 colors but not for flatness of field. Many are used for photography but are appropriately cropped to exclude the unfocused areas.
The cream of the crop are the Planapochromats (or planapos) and these are corrected for everything but unfortunately are subject to delamination, at least for some brands. So, eBay buyer beware. These lenses also enjoy the highest NA figures so are best if resolution is a primary goal.
And what type of microscope should you be aiming these objectives for, given that you want to study pond water? Least attractive is brightfield but with oblique illumination you can enhance contrast quite a bit, perhaps enough to enjoy the hobby. But in my mind the minimum you want to look at is Phase contrast and if you can afford it...DIC is the gift from the gods when it comes to viewing and photographing phase objects.
That's it for now. Hopefully I haven't lead you astray and this little explanation will lead to further study. After these basic considerations there is the whole subject of how they are to be used, lit, aligned, etc. Google can again be your friend.