The central dogma of biology says that DNA encodes RNA which encodes proteins
In the beginning you were a single cell made up from a fused sperm and egg. This cell had 23 pairs of chromosomes which together made up all your DNA - chromosomes are what DNA looks like when it's tightly wrapped. DNA is a long code made up of four nucleotides that form a four letter alphabet - ACGT. This four letter chemical alphabet contains every single instruction the cell needs. The most important thing it does is code for proteins. But the DNA code also encodes the information for how it should be read, how it should coil up, and which things in the cell it should interact with: Life is a self-executing program.
DNA does not make proteins directly. Instead, when a protein needs to be made then the DNA is transcripted into RNA. RNA is similar to DNA, but it uses the letter U in place of T. The RNA can then be read by other special proteins which translate the four letter RNA code (ACGU) into the twenty letter amino acid code. The RNA code is read as an amino acid code, and then those amino acids are connected together to form a larger molecule which is the protein.
Easy: DNA is transcripted into RNA which is translated into proteins.
Cells decide what proteins they need and then get the protein's code from the DNA
Let's imagine DNA as a long line of letters sitting in the middle of a cell. Maybe this is an immune cell - a cell that helps attack pathogens and other potential threats. Something's happened and the cell's decided it needs to respond. A protein, or groups of proteins, will be recruited. They will be recruited by having their shapes change or having some small molecules attached to them. When these molecules get near the DNA they can attach to the DNA.
As I said above, DNA doesn't just encode proteins, it also encodes its own shape. The actual letters that are the DNA affect the way the DNA coils. It also affects the way proteins can dock on it. Some sequences of letters are specifically written to encourage particular proteins to dock to them - the proteins themselves can dock because they themselves have the right amino acids in the right shape to allow for docking.
Continuing the example: the immune cell has sent some proteins that can dock to important parts of the DNA. Once those proteins have docked they can affect the way the DNA transcripts. They might do this by changing the shape of the DNA - perhaps they force a kink in the DNA, so that some parts are hidden and others more available; or perhaps they act as docks of their own, encouraging proteins which are responsible for transcription to approach the site. There are myriad ways for transcription to be affected by proteins.
DNA contains instructions which need to be taken elsewhere as RNA to be enacted
Let's say the protein acts as a dock. In this example the docking protein helps a special class of protein which is responsible for reading/transcripting the DNA. This protein complex is called RNA polymerase. It attaches to the DNA, looks for nearby letters which say "START here", and then keeps reading until it reaches the letters which correspond to "END here". Between those two spots, the RNA polymerase copies the DNA into a message form. This message form is very similar to the original DNA, but is now RNA, and is a short snippet of code which can be sent off to be made into actual proteins.
In this example an immune cell sent a protein to act as a dock on the DNA. The protein helped another protein called RNA polymerase attach. The RNA polymerase copied out the relevant part of DNA into RNA. This is a type of RNA called messenger RNA (mRNA). It's a single instruction which can now be carried off, away from the DNA, to be put into action to make a protein.
Different parts of the DNA are controlled differently. It's important that only the instructions a cell needs at the moment are put into action. Terminology-wise, a part of a DNA which corresponds to making a single protein can be thought of as a single instruction, and is also called a gene. Once a DNA instruction, or gene, has been copied into mRNA that mRNA is sent off to be made into proteins. Some genes are always on, but their activity is modified with need, while others are turned off completely, and others are somewhere in between. If a cell can't control what instructions are made then it will make the wrong proteins in the wrong amounts and will probably die or cause damage to its host organism.
Translation and off to work
Once a gene has been copied into mRNA it's ready to be made into proteins. This involves the mRNA being read by another series of proteins, called a ribosome. The ribosome latches onto the beginning of the mRNA string and scrolls through it until it reaches the end. The RNA codes for amino acids; different combinations of RNA letters tell the ribosome which amino acid to attach next. In this way the protein is made by attaching one amino acid at a time.
Take a common protein - actin, which forms structural supports in the cell - back of the napkin calculations suggest that there's something like a million of these actin proteins in a single cell*. Now take into account that there are around 23,000 genes in humans - many of which can be cut and pasted to form unique variants, and you begin to fathom the scale of the problem. Add to that the fact that each of these proteins' transcription, translation, and final recycling are controlled by multiple mechanisms, that proteins can undergo a huge number of modifications once they've been made, that they can be shuttled off to specific corners of the cell, and that they each interact with multiple other proteins to do all sorts of different things and you can imagine that most biologists will only be aware of a small fraction of all proteins, and very familiar with even less.
But why are proteins made? It's useful to think of the different jobs that proteins do. Obviously only small list can be given here, but it'll suggest their varieties. Some proteins form structural elements, some let specific atoms which are needed into the cell, some are sent out of the cell to grab onto bacteria, some are receivers which sit on the outside of neurons waiting for signals, some integrate signals from outside the cell and decide whether the cell should grow, and some act in long chemical synthesis arrays, making everything from fats and sugars onwards. Everything that is us in our body is either a product of DNA, especially proteins, or else molecules which have been made by proteins. We get new molecules from food which we've digested (thanks to proteins) and then integrated into our cell (thanks to proteins).
Proteins are what get things done and their ability to be encoded by DNA and then made as needed is the basis for life.
* This was calculated as follows: If actin is present at 63E-6M in yeast (+) and if the volume of a cell is around 29E-15L (+), then there's around 1.8E-18mol of actin per cell. Multiply that by the Avogadro constant and you get 1.1E6 or 1.1 million individual actin proteins per cell. Obviously this number varies MASSIVELY depending on pretty much anything.