Put most simply, a cell is the basic unit of a cellular system, which is kind of backward, since its saying that a cell is part of something made out of cells! Another way of looking at it is as though a cell were a brick, and the body is an enormous wall or a house - made up of cells. Unfortunately this falls down again, because a brick is a relatively simple structure, whereas a cell can be very complicated!
The image represents what could be described as a typical eukaryotic cell, containing the basic features of all cells.
When it is made, every human cell contains its own living instructions, which it received from the cell that it came from - kind of like having a bit of your parents hanging about with you all the time, telling you what to do. This is in the form of DNA that can be found in the nucleus.
A prokaryote is a kind of really simple cell, because it doesn't have a nucleus, or many of the other membrane-bound organelles which you'll usually find in a cell.
In fact, prokaryotes are usually unicellular - i.e. they exist as just one cell on their own - unlike eukaryotes, which are sometimes unicellular, but often multicellular.
They still have to perform certain functions - including producing proteins - so there are certain key features:
A nucleoid region, containing the DNA needed to make proteins, but importantly not arranged in a nucleus;
Ribosomes, which is where the process of translation takes place;
A membrane, to keep everything together;
There are also features which are exclusive to prokaryotes.
The most important piece of information when considering prokaryotes is that they don't have the membrane-bound organelles you find in eukaryotes. The word pro-karyote literally means 'before the nucleus', and if you subscribe to the theory of evolution, the suggestion is that prokaryotes came about before the nucleus had 'evolved'.
It is generally considered that prokaryotes are bacteria, and that's about it - in actual fact there are other organisms involved, but bacteria make up the greatest proportion of prokaryotic cells.
Eu-karyote means 'true-nucleus', to the extent that, unlike prokaryotes, eukaryotic cells contain a nucleus. They also contain many other organelles which are not found in the prokaryote.
Eukaryotes can be either multicellular or unicellular - that is, the total number of cells in a particular eukaryote can vary. Humans have millions of cells, because we are relatively large organisms, but there are organisms which are much smaller.
The important feature of a eukaryote is the nucleus which you will invariably find in its cells (though of course humans are eukaryotes, yet their mature red blood cells do not contain a nucleus!) and this is a very important distinction to make.
A typical eukaryotic cell will contain a whole range of organelles, including the ribosomes and membrane found in prokaryotes. They will also have mitochondria, which produce energy in the form of ATP; rough endoplasmic reticulum, which - coated in ribosomes as they are - help to produce proteins; smooth endoplasmic reticulum, which produce certain types of hormones; vesicles, which are involved in import and export of substances; and more!
Importantly eukaryotes have a nucleus and other organelles. They are the cells which you find in plants and animals (including humans!) and their complexity reflects an incredibly efficient, well-structured design.