All plants are Eukaryota. Algae are a kind of plant. All algae are
eukaryota. Algae are distinguished from other plants by their lack of
cellular differentiation.

All algae have chloroplasts. All plants have chloroplasts, which are
captured bacterial symbionts. The capture event is over a billion years
back, if rRNA is any good as a molecular clock. (So far, so good, at
least) Unlike most other organelles in plant cells, chloroplasts have
their own DNA and control their own division and replication apart from
their host.

Unfortunately, some people off handedly call cyanobacteria, green
sulfur, green non-sulfur, purple non-sulfur, and purple sulfur
photosynthetic bacteria algae when they're not- they're Prokayota,
commonly refered to as bacteria.


Further reading on the web:
Explanation of celluar differentiation:
http://en.wikipedia.org/wiki/Cellular_differentiation
Lynn Margulis' endosymbiont theory:
http://www.pnas.org/cgi/content/abstract/93/3/1071
The Tree of Life:
http://tolweb.org/tree/phylogeny.html
The Ribosomal Batabase Project
http://rdp.cme.msu.edu/index.jsp