Phylum Cyanophyta - The Blue Greens
Only phyla that are prokaryotic (bacterial) cells, chlorophylls A and phycobilins (auxiliary pigments), reproduction is usually binary fission, occasionally.  Cells vary from single cells to aggregates to filaments.  Fix atmospheric nitrogen adding it into the food chain.  In polluted waters it can deplete available oxygen which is detrimental to animal life (i.e. fish kills).
   Order Chroococcales -solitary cells, aggregations and colonies
        Family Chroococcaceae - Gloeocapsa, Merismopedia, Microcystis
   Order Oscillatoriales - filamentous without heterocysts
        Family Oscillatoriaceae - Oscillatoria, Lyngbya, Phormidum, Spirulina
       Family Stigonemataceae - Stigonema (Fischerella)
   Order Nostocales - filaments with heterocysts
        Family Nostocaceae (unbranched)  - Anabaena, Cylindrospermum,  Nostoc
       Family Scytonemataceae (false branching) filaments with heterocysts - Scytonema
        Family Rivulariaceae - Calothrix, Rivularia

Phylum Chlorophyta - The Greens
Chlorophylls A and B, starch carbohydrate reserve.  Cells vary from single celled (Chlamydomonas sp.) to colonies (Volvox sp.) to filamentous (Oogonium sp.) to complex branched structures (Chara sp.).  Is theorized to be an ancestor to vascular and non-vascular plants.  Important food source for animals and  people.
Subphylum Chlorophyceae
   Order Chlorococcales - solitary or aggregations or colonies, all non-motile in vegetative
   state
        Family Chlorococcales - non flagellated single celled algae that have a motile cycle
        during reproduction (planospores) and a non-motile cycle (aplanospores)
        Chlorococcum, Closteridium, Trebouxia
        Family Oocystaceae - elongated, fusiform, non flagellated single cell algae -
        Ankistrodesmus
       Family Scenedesmaceae - colonial cells in groups of 2 to 4, may bear spines -
        Scenedesmus
        Family Hydrodictaceae - organized colonial cells forming - Hydrodictyon, Pediastrum
    Order Chaetophorales - filamentous algae with differentiation in growing patterns
    (heterotrichy)
        Family Chaetopharaceae - Draparnaldia, Stigeoclonium
        Family Coleochaetaceae - Coleochaete
    Order Cladophora - multicellular algae that are coenocytic, filamentous
        Family Cladophoraceae - Cladophora
   Order Oedogoniales -filaments are branched / unbranched but without
    differentiation in growing patterns, filaments are specialized for reproduction:
    complex oogonium formed during reproduction
        Family Oedogoniaceae - Bulbochaete, Oedogonium
   Order Tetrasporales - non flagellated cells aggregated in mucilaginous colonies
        Family Gloecocystis - colonies forming "sheets" of cells in mucilage - Gloeocystis
   Order Ulotrichales - unbranced, filamentous algae
        Family Ulotrichaceae - Ulothrix
        Family Microsporaceae - Microspora
   Order Volvocales - have flagellated cells and colonies with glycoprotein cell walls
        Family Chlamydomonadaceae - Solitary flagellated cells - Chlamydomonas
        Family Volvocaceae - colonial, flagellated cells - Eudorina , Gonium, Pandorina ,
        Volvox
    Order Zygnematales (Conjugales) - distinguished because uses conjugation to
    exchange genetic material during when undergoing sexual reproduction, may be
    filamentous or unicellular
       Family Zygnemataceae - Mougeotia, Spirogyra, Zygnema
       Family Desmidiaceae - Closterium, Cosmarium, Desmidium, Micrasterias, Staurastrum
Subphyla Charophyceae
   Order Charales -macrophytic algae with coenocytic cells
         Family Characeae - Chara, Nitella

Phylum Euglenophyta - The Euglenoids
Chlorophylls A and B, paramylon carbohydrate reserve.  Single cells with one or two flagella.
    Order Euglenales - have flagellated cells and colonies with glycoprotein cell walls.
        Family Euglenaceae - Euglena, Phacus, Trachelomonas

Phylum Pyrrhophyta - The Dinoflagellates
Chlorophylls A and C, starch carbohydrate reserve, cells covered with a theca.  Have two flagella.
Important phytoplankton component and food source for other organisms.  Can also have negative effects on animal life as in the case of red tides.
    Order Dinokontae - lateral flagella, one is ribbon-like and encircles the cells, the other
    flagella extends out from the cell.
        Family Peridiniaceae - Peridinium

Phylum Chrysophyta - The Golden Browns, Yellow-Greens, Diatoms
Chlorophylls A and C, chrysolaminarin carbohydrate reserve, flagellated in at least part of their life cycle.
   Subphylum Xanthophyceae - Yellow Green algae, usually pale-green in color,
    chlorophyll A and  C, cell walls are pectin or cellulose
   Order Mischococcales - elongated, cylindrical cells attached at least at one end
        Family Sciadaceae - Ophiocytium
    Order Tribonematales - Filamentous, lacking branches
        Family Xanthophyceae - Tribonema
    Order Vaucheriales - coenocytic, filamentous
        Family Vaucheriaceae - Vaucheria
    Subphylum Chrysophyceae - Golden Brown Algae, chlorophylls A, C and fucoxanthins
    (auxiliary pigments), cell walls are scales
        Order Ochromonadaceae - colonies of flagellated cells
            Family Dinobryaceae - Dinobryon
            Family Synuraceae - Synura
Subphylum Bacillariophyceae (Diatomaceae) - Diatoms, chlorophylls A, C and fucoxanthins  (auxiliary pigments), cell walls composed of silica, classification based on ornate cell wall markings and their shape and symmetry.  Silica in cell walls is used in many abrasive compounds as well as diatomaceous earth.
    Order Centrales - radially symmetric diatoms
        Family Coscinodiscaceae - Cyclotella, Melosira, Stephanodiscus
    Order Pennales - bilaterally symmetric diatoms
        Family Cymbellaceae - Cymbella
        Family Fragilariaceae - Asterionella, Diatoma, Fragilaria, Synedra, Tabellaria
        Family Naviculaceae - Diatomella, Navicula, Gyrosigma
        Family Gomphonemaceae - Gomphonema
        Family Cymbellaceae - Cymbella
        Family Nitzschiaceae - Bacillaria, Nitzschia

Phylum Rhodophyta - The Reds
Chlorophylls A and phycobilins (auxiliary pigments), color is based on proportion of phycobilins (red is produced by phycoerythrin) no flagellated stages at all.  Cell walls may be cellulose but sometimes other sugar polymers (such as mannose / xylose polymers).  Have mucus-like sugars (called mucopolysaccharides).  Some of these are what we derive food thickness such as agar and carrageenan.   The members tend to be filamentous and some are quiet complex but single-celled organisms are found.
    Order Bangiales - unbranched filamentous algae
        Family Goniotrachaceae - Asterocystis, Porphyridium
        Family  Bangiaceae - Bangia
    Order Nemalionales - branched filamentous algae with differentiation in growing
    patterns (heterotrichy),  basal cell attached to a substrate.
        Family Batrachospermaceae - more specialized filaments Batrachospermum
        Family Chantransiaceae - less specialized filaments Audouinella