@wmd Plankton is an umbrella term for a variety of mostly microscopic ocean life. It includes all kinds of things from bacteria and archaea to tiny arthropods to larval forms of other sea creatures. Weirdly, a few larger things are technically plankton too, like jellyfish!
Algae is a similarly informal term with no particular definition. It covers a variety of photosynthetic eukaryotic organisms, some single celled, some more complex (like kelp). Most algae are plants, but some protists are also considered algae. Cyanobacteria are usually excluded from this group though, even though they're colloquially called "blue-green algae".
I'm pretty sure these are some old terms, which is why they don't really match modern classifications.