About algae. "Algae" is a convenient "Other" category that resulted when early biologists threw together as "plants" all the photosynthesizing organisms that made their own food from light, and then removed all the mosses and liverworts and "higher" vascular plants, calling the remainder "algae." In that residue were lumped together various photosynthesizers that had no common genealogy, no shared evolutionary history. But "algae" is still a useful working category, like "invertebrates," that other familiar miscellaneous "everything else" pigeonhole. The internet AlgaeBase covers blue-green algae (cyanobacteria), rhodophytes, phaeophytes, chlorophytes, euglenophytes, charophytes, diatoms, dinoflagellates and cryptophytes, together with a number of smaller groups.
Algae lack roots, stems and leaves and the "vascular" tissue that channels nutrients from one kind of tissue to another, but their simple thalli, which may exist as individual cells, can form filaments and other colonial structures.Algae lie at the base of the food web, providing nutrition and shelter for the smallest planktonic grazers in the open water or in the biofilm. The algae are many, but, by the way, a single one is an "alga" (with a hard g, like "Olga"). Saying "algaes" is like saying "feetses."
Biologists identify and set apart the broadest algal groups partly by the characteristic sets of photosynthetic pigment types each group has in common. Algae use an array of photosynthesizing pigments that are much more various than the narrow selection the "higher" plants have inherited from their green algal ancestors. The various chlorophylls are the energy-trapping pigments we're most familiar with, chlorophyll a, the universally-used pigment that powers most photosynthetic chemistry; but there are other forms of chlorophyll and other, auxiliary photosynthetic pigments, like carotenoids: all the freshwater algae produce beta-carotene. Most also produce some auxiliary photosynthesizing yellow pigments, xanthophylls. The green algae (chlorophytes) use chlorophyll b in addition to a. The completely unrelated red algae (rhodophytes) use no chlorophyll b but manufacture instead some chlorophyll c in addition to their chlorophyll a and another chromophore, phycobilin, which they share only with the cyanobacteria. So those rhodophytes are as distinct in their metabolism from green algae — and most probably as unrelated in their genetic history, too — as a sequoia is from the fungus that grows on its trunk. Aside from the universal chlorophyll a, there's no overlap in the photochemistries that characterize each group... and probably no shared family history, either.
So the algae vary in their most basic biochemistry and in the compounds they use for building cell walls and storing energy. Conveniently, though, the various morphological types of algae divide up pretty much the same way as their differing photosynthetic apparatus suggests.
Why should such biology trivia be of interest to you? Well, for one thing these various chlorophylls each have a slightly different absorption spectrum, the range of light wavelengths that they use. So it strikes me that, if one of your tanks has chronic trouble with black brush algae, say, or with cyanobacteria, a fresh fluorescent bulb with a slightly different spectrum, or a switch to LED lighting, might be just enough to tip the scales in favor of the higher plants. Or, fishkeepers have found, sometimes all it takes is switching the lighting in its reflectors between two tanks, without changing any tubes, to set back the algae in each tank.
If you want to outwit your enemy, it pays to know your enemy. And now you know algae's dirty secret: algae are not a related group of organisms after all, just a convenient designation for photosynthesizing eucaryotes with cellulose cell walls, whether they are single-celled or form colonial filaments, sheets or spheres, or even the huge multicellular structures (called "thalli") of kelp.
Links. "The varieties of algae" at Micrographia might be a good place to start; the beauty of the microscopic illustrations for which Micrographia is outstanding will give you a fresh perspective on what you're scraping off the glass.
The kinds of freshwater algae, of interest to the fish-keeper as much as the pond-dipper, are set out in the form of a table at Microscopy-UK, compiled by Wim van Egmond and Dave Walker.
The Algal Web is an aid to identifying algae under the microscope, but it also offers checklists of algae found in particular habitats.
The University of Carolina's site Algae Groups offers more information on each grouping. The Freshwater Ecology Laboratory at Connecticut College generates the Silica Secchi Disk algal site, with many components.