How does the size of our cells compare with other organisms?

I can't do better than the source below, so I'm simply going to quote it, with added italics:

...cells, from whatever organisms are of quite similar size. Most bacterial cells are about 1-2 μm in diameter, and most cells of higher organisms are only about 10-20 times larger, plant cells being somewhat larger than animal cells. There are, to be sure, exceptions: There are very small bacteria (0.2 μm), and there are unusual cells like those in the nervous systems of vertebrates, some of which may be over 1 m long. But compared with the overall range of sizes of natural organisms, all cell sizes are much alike.

In both plants and animals, the size of the cells bears no relationship to the size of the organism. An elephant and a flea have cells of about the same size; the elephant just has more of them. Why is such uniformity in cell size maintained? A clue can be found in the fact that the surface area/volume ratio for an object of a given shape depends on its size. The complex chemical processes in a cell and the large molecules that participate in them require a significant volume. Yet the cell must also exchange substances with its surroundings to support the active metabolism within. Too large a cell will not have enough surface for this exchange to occur, unless it is highly elongated like a vertebrate nerve cell, increasing the surface/volume ratio. Bacterial cells are smaller than the cells of higher organisms because bacterial metabolism is simpler. Viruses, which are even smaller than bacteria, do not have a metabolism of their own but exist as parasites in the cells they invade.

The only thing I'd add would be to clarify the surface area/volume ratio issue. Imagine two cubes (only because the maths is easier - the principle is the same). One has sides of 1 μm, and the other's sides measure 10 μm.

First let us compare surface areas. Each cube has six faces. The first cube must therefore have a surface area of 6 x (1 x 1 μm2), or 6 μm2. The second comfortably dwarfs it competitor, coming in at 6 x (10 x 10 μm2), 0r 600 μm2.

Next, compare volume. The first cube's maths are easy - 1 x 1 x 1 = 1 μm3. The second one's volume equates to 10 x 10 x 10 = 1000 μm3.

So the second has both a greater surface area and volume, obviously. But now compare the ratio of the two measurements. The smaller cube's surface area/volume ratio is 6:1, whereas the same calculation for the second cube comes out as 0.6. Suddenly, the smaller cube comes from nowhere to win the race.

The moral of the story? Smaller objects have a greater surface area to volume ratio; bigger objects a smaller one. Clearly, for any given cell there is an optimum ratio between surface area to exchanging things with its environment (like oxygen, wastes and glucose) and volume to fit as much cellular machinery in as possible. Each parameter 'wants' to be as large as possible, but unfortunately they can't vary independently. Changing one must change the other too, and a cell has to find a suitable compromise.

So cell size doesn't vary capriciously. For similar types of cells (e.g. vertebrate cells), the size is kept remarkably constant, with very few exceptions.