To give you a rough idea: DNA carries information, like the letters you are reading right now. In our cells there is machinery that translates this information to RNA - a different type of long chain - and then most often this gets further translated to proteins - another chain of molecules. Most interesting things in cells happen because of proteins, because they can form enzymes, little machines that can do one task. The same way you put bread in your toaster and toasted bread comes out you can put one type of molecule (or two, or several) into an enzyme and it will do something with it.

The information on DNA can be simplified like a sort of machine code: it looks something like this:

blablablabla->PROTEIN1||blablabla

In my made-up code blabla is just some irrelevant code, the arrow signifies a start signal and the || means stop. The little machine reads the DNA sequence and starts to translate when it reads a start sign and stops to translate when it reads a ||. This way protein 1 is made. On your DNA there are about 30.000 of those proteins than your cells can potentially make.

But not every cell produces all the same proteins and not in the same amount and here we come to the actual answer to your question: certain proteins can change the code by either blocking the start signal - a gene is "switched off" - or can make the machine read the same command several times, making the signal stronger. And this is the way regulation in your cells work: Cells produce proteins that switch on or off or enhance or weaken the expression - the reading and translating - of certain genes. So all regulations starts with your first starting cell - the so called zygote - that has some basic genes turned on and gets information from your mother during your growth. After that it's an unbroken chain of cells making proteins that tell other cells to make certain proteins and so on. One cell makes proteins that produce an attachement on one side and a lot of little flaps on the other because it got "told" by its neighbors to do so because it is in your gut. Each tissue has these informations by sending out certain signals that turn on and off genes in newly formed cells.

Hope that helps a bit.

Now to what all of this has to do with different cell types: cell types are in the end only determined by the slightly different combinations of proteins, fats and carbohydrates they produce and arrage in and on themselves. When you were first formed your original cell (the so called zygote) only had information to duplicate itself. Your mother gave it some information on directions - what is up, down, left or right. Once the directions are established more tricky things are achieved by sequences of commands: first these cells form a huge ball but then the cells that point in one direction get the information to start different genes in your DNA