What kind of living things are there? What kind are we?
With some qualifications we need to address in a minute, living things come in two kinds: bacteria (informally) and eukaryotes. Bacteria are always single cells, and eukaryotes are mostly single cells. The exceptions are things like you and me and rose bushes and hyenas and leeches and kelp, which are big, multi-celled bodies. We are eukaryotes. All living things have a cell membrane, which keeps the DNA separate from the ocean. The basic difference at the topmost level of life between bacteria and eukaryotes is that we have a second membrane, inside the first, and our DNA is inside it: this is the nucleus, and that gives the group its name (Greek eu "well, good", karyon "nut" i.e. nucleus).
A historical recap
It turns out this is pretty much the fundamental difference between the two kinds. It wasn't always so: the traditional idea (certainly in Linnaeus, and probably going back to Aristotle) was that some folk lay around and basked in the sun, and some folk moved hither and thither and et the other folk. These were called respectively plants and animals. Linnaeus put two so-called kingdoms at the top of his scientific classification, Animalia and Plantae.
The big difference between Aristotle and Linnaeus is that in the intervening 2000-odd years the microscope had revealed all sorts of weird living things called, quite generally, microbes. At first it seemed the Aristotelean dichotomy still worked: some of them basked, and some of them ate. (Later we discovered photosynthesis: so that was why they could just bask in the sun. Later still we discovered some others that could bask in the sulfur in vents deep in the sea.) But at first, you could assign all the tiny little beasties to being either protozoa (titchy little animals, Greek zoon "animal"), or algae (titchy little plants, Latin alga "seaweed").
Move forward from Linnaeus into the nineteenth century and a bit beyond. Microscopes become more powerful. The cell is discovered. The germ theory of disease is put forward by Pasteur. We discovered photosynthesis, bacteria, and viruses. (We'll have to mention viruses when we have that chat about 'qualifications' I promised earlier.) You might still get mushrooms in the vegetable section of your local supermarket, but there were now scientists peering down microscopes at fungal cell walls, muttering, 'No way that is a plant.' So what is it then?
The kinds of eukaryotes
The early efforts, which held sway until very recently, was that there were Animals and there were Plants and there were one or two or three other kingdoms too (culminating in The Five Kingdoms): the names Protista and Protoctista and prokaryote came in, possibly from the Greek for "all those other things we're not sure about", or possibly not. Then came DNA sequencing.
From about 1990 biologists could actually see the ancestry of all living things. Whereas classification by morphology compared tens or hundreds of features, DNA sequences allow the comparison of tens or hundreds of thousands of gradual blips in the evolutionary sequence. The early results were chaotic: instead of big dominant kingdoms of animals and plants being joined by a few others, it turned out animals and plants were only leaves at the end of a very tangled (and still unresolved) tree of eukaryotic life, and perhaps 20 or 30 or 50 kingdoms might be called for. This is unsatisfactory. We need to reduce this by grouping them into superkingdoms. After a while it became apparent by about 2000 that most eukaryotes could be put into some seven major groups: opisthokonts (that's us animals and, surprisingly, fungi); Amoebozoa; Archaeplastida (that's mainly plants); Chromista; alveolates; Rhizaria; and excavates.
The problem with describing these is that most of us have no idea what most of these are. They either live in the sea like diatoms and various kinds of plankton, or they cause various parasitic diseases, or they're just there underneath rocks but too small to see . . . but I have no real feel for ciliates, Euglena, parabasalids, Oomycetes, Giardia, vahlkampfiid amoebae, cryptomonads, or any of the other things that appear on the family tree. But these, not mere 'plants and animals', are the real diversity of eukaryotic life. Also, as you might have noticed, I'm flicking backwards and forwards between capitalized taxon names like Euglena and less formal plural group names like ciliates. This reflects the fact that most of the scientists actually working in this area aren't really bothered whether it's a kingdom, subkingdom, or phylum, or what Classical suffix the name should end in (or whether we're now talking about Eucarya, Eukarya, or Eukaryota). What's important is the family tree, the phylogeny. Each clade derives from some earlier clade: those are the facts of its history and the truth of its classification.
In more recent years the groupings could be made a bit tighter. For a start, opisthokonts and Amoebozoa belong together as unikonts. (Amoebozoa contains our old friend Amoeba and many but not all things that have an amoeba-like shape and lifestyle.) Chromists and alveolates belong together as chromalveolates. So we're down to five: Unikonts, Archaeplastids (called Plantae by some), Chromalveolates, Excavates, and Rhizaria. And five times evolution has created large, permanent multi-celled organisms, but they don't match these five subdomains: rather, animals and fungi are unikonts, green plants and seaweeds and Rhodophyta (red seaweed) are archaeplastids, and Phaeophyta (brown seaweed) are chromalveolates; all islands of multicellularity in a sea of tiny diatoms and ciliates and little-known algae. (Some other groups have small or temporary multicellular forms, several spending some of their time as 'slime moulds'; the small colonial marine plant Volvox has differentiated sex and body cells, a separate development from most multi-celled plants.)
This is the situation as I learnt it not many years ago. In researching this node I found more progress has been made: it now seems that the basic difference is between unikonts and bikonts, with bikonts covering the other four, but Rhizaria may now be within Chromalveolates (or something similar called the SAR clade) rather than parallel. There's no point being more than vague about this at this time. Every single node on E2 about phylogeny at this level is out of date, but there's nothing wrong with that. This node will be too, soon enough. All I want to do is give an impression of the current state of thinking. I recommend a Google Images search for "eukaryote phylogeny": you'll see these tumbleweeds of doubtful branching, with thinking changing over the last fifteen years or so, and nothing completely agreed upon yet.
The four kinds of life
Living things don't come in two kinds. First there are Archaea. These look like bacteria, until you train a microscope on certain fundamental aspects of their construction. They're typically extremophiles, living in intense heat or cold or salt. The Three Domains theory of Carl Woese divided life up into Bacteria, Archaea, and eukaryotes, and this is now accepted. It pre-dated the flood of DNA evidence. We have no real idea whether their common ancestor was more like a bacterium or an archaean. We know it can't have been a eukaryote, for as we shall see below, eukaryotes are hybrids of the others. Some group Archaea and eukaryotes together as Neomura.
Second, what about viruses? The answer is we just don't know enough about how they're related to cellular life or to each other. Are they degenerate organisms that settled into parasitism and ditched the DNA they didn't need? Are they chunks of chromosome or plasmid that ventured into the outside world? (Are they all a single group?) Many scientists avoid this question by saying they're not living things, as they can't survive and reproduce independently. Well, a jumping gene or selfish DNA might be like a string of beads with an Alice in Wonderland label 'COPY ME', but a virus is more like an A4 sheet with the words 'COPY ME' above a plan of your office with the photocopier marked with a red X, and wearing a black mask and carrying a jemmy and a glass-cutter. And some viruses, like the mimivirus group, are huge, comparable to bacteria, or to that plan of your office held by Barney from Mission: Impossible driving the Mole from Thunderbirds. So maybe some day they'll be a fourth domain.
Eat your blue-greens and reds
The very idea of a genetic family tree is questionable at this level, because of horizontal gene transfer (prokaryotes are quite happy to swap bits of DNA) and endosymbiosis. The latter is when one organism ate or otherwise engulfed another and failed to digest it; the inner organism's genes and cell membrane survived intact and they now form one fused cell, which propagates its kind. This has happened in at least three major events in the history of eukaryotes. First, the lineage was apparently created when some archaean ingested some α-proteobacterium: these survive as the mitochondria, which supply us with energy. A few weirdos have secondarily lost them; otherwise we all have mitochondria. Then, presumably, came the division between unikonts and bikonts. The second endosymbiotic event was when some primordial bikont ingested a cyanobacterium, which already knew how to do photosynthesis. This undigested meal became another organelle, the chloroplast, the most basic kind of plastid: and thus were created the archaeplastids (plants, in the broadest sense). Not all plastids have the same number of cell membranes. It now seems chromalveolates came about when their ancestor swallowed an archaeplastid, specifically a rhodophyte or red alga. Again, the phylogenetic situation is confused, because descendants sometimes dispense with these bits.
What I've tried to give is a brief and untechnical overview of the kinds of life and how they're related. The perceived facts will change rapidly as better DNA evidence comes in. A more complete description of many of these groups would just blur into a list of names of things most of us have never heard of, but I think it is useful and salutary to see where we fit into the big picture. Other nodes here, and the big wide world outside E2, give more technical detail if you want it.
Life is bacteria, archaeans, and eukaryotes, and possibly viruses. Eukaryotes are bacteria/archaean hybrids, and come as unikonts or bikonts. Unikonts are principally animals, fungi, and amoebae. Bikonts are plants, chromalveolates, rhizaria, and excavates. Plants come from a second endosymbiosis, chromalveolates from a third.