Your question relates to the similarities an differences that we see between animal and plant cells. Both are, of course, "eukaryotes," and so much of the cellular structures, organelles, and machinery are quite similar.

As you probably know, the centrioles are fascinating, cylindrical structures embedded in the organelle called the "Centrosome." The centrioles are made of polymers of tubulin (actually, a specialized type of tubulin called gamma tubulin) protein and lots(perhaps hundreds) of accessory proteins arranged at right angles to one another, forming a sort of L-shape. The centrioles in animal cells organize microtubules, especially to form the mitotic spindle for cell division. In motile cells, they also give rise to the basal bodies of cilia and flagella. The centrioles are essential for the faithful duplication of the centrosome (especially the "matrix" or pericentriolar material of the centrosome) during cell division, and the centrioles themselves duplicate as well. We do not know much about the mechanism of this process, however.

Now to your question - it turns out that while all eukaryotic cells have some sort of ""microtubule organizing center (MTOC)" or centrosome, neither fungi, lower plants (alagae, diatoms), nor MOST higher plant cells contain centrioles. In higher plants, cells seem to nucleate microtubules at sites distributed all around the nuclear envelope. However, they do use the special tubulin (gamma tubulin) to nucleate microtubules, just like the centrioles do in animal cells. There are a few examples of plant cells that appear to have a structure that looks similar to an animal cell centrioles.

You might think about how the structure of plant cells differs from that of animal cells and how this might affect cell division processes. It is also interesting to think about how the same protein (gamma tubulin) can be used to do the same job (nucleate or "organize" microtubules) in different cells yet use very different mechanisms to do so. We don't know why an animal cell uses the complex centriole embedded in an even more complex centrosome while a higher plant cell there does not seem to be single, coordinated MTOC.

Yes - only bacteria and some amoebas lack centriols (actually, I don't think dinoflagelates - a type of microscopic algae - have centrioles either, but I would need to check).

What plant cells don't have is the same microstructure that maintains and furrows the cell membrane during cell division. Instead, they have a cellulose cell wall, and instead of breaking up and dividing to separate the cytoplasms, they make a rigid plate of cellulose between where the new cells are to be.