Simplicity is supposed to be genius so let's try and keep this simple - particularly for you folks out there who are amateur Biologists with a keen interest in the subject - good for you!
Living things are made of cells - there are two main types of cells - prokaryotic and eukaryotic. Prokaryotic cells are what the common man would call bacteria and eukaryotic cells are plant, animal and fungi cells.
The main difference between eukaryotic and prokaryotic cells lies in how the nuclear material (i.e DNA) is organised in the cell. Eukaryotes have the DNA enclosed in a nucleus - surrounded by a double membrane with pores in it to allow communication with the cell cytoplasm. Prokaryotes on the other hand have their DNA floating free in the cytoplasm - it often takes the form of a single, wound up chromosome and possibly some small circular pieces called 'plasmids' as well.
In fact, the lack of any double membrane surrounded organelles (structures like the nucleus and mitochondria) is what characterises and summarises the differences between prokaryotic and eukaryotic cells.
The lack of these organelles, then, indicates a less complex structure of the prokaryotic cell in comparison to eukaryotic cells.
So, instead of performing respiration reactions within mitochondria like a eukaryote the prokaryote uses the folded cell membrane as its respiratory membrane. Eukaryotes have ribosomes that are organised onto the membrane bound chambers of the rough endoplasmic reticulum while prokaryotes have their ribosomes free in the cytoplasm.
No surprise then that evolutionary theory suggest the appearance of prokaryotes on the earth before that of eukaryotes. Interestingly, mitochondria and chloroplasts actually contain their own DNA in the form of plasmids and actually resemble prokaryotes as well, in that mitochondria nd chloroplasts also have folded membranes and their own ribosomes.
So, what may have happened to cause eukaryotes to evolve was the invasion of a cell by prokaryotes that took up residence in a symbiotic (mutually beneficial) relationship with the cell.
Prokaryotes are unicellular and divide rapidly by mitosis (simple cell division to create two identical daughter cells). They may however connect (called conjugation) and exchange and recombine their DNA. Thus bacteria can become multiple resistant to antibiotics through mutation, rapid division and conjugation.
Prokaryotes cause diseases such as cholera and tuberculosis which afflict millions around the world. However genetically engineered bacteria that have had human genes inserted produce insulin for diabetics, human growth hormone and factor eight for haemophiliacs.
The greater our understanding of these tiny cells (they are usually about a hundredth of the size of a eukaryotic cell) then the better it will be for the human race.