The first step to do is to make sure your version of Minecraft is up to date, and make sure that the other person your playing with is also on the same version or else they will not be able to join your LAN game. For this example we will be using Release 1.7.9
On the main menu of Minecraft, click the Single player option to access the world creation screen.
Click the button "Create new world" to go into the world creation process.
On this screen, you can name your world, for this example I chose something completely random, and also I have my Game Mode set to Creative Mode.
proceed to click Create New World, and your world will begin to generate.
This process doesn't take too long, but can take long depending on certain optional selection you can make in world creation menus.
And here we are! Our world has been generated and we are playing in creative mode.
Process to press Escape on your keyboard, and on this screen click "Open to LAN".
On this screen you can select the settings for your other players, you can change their gamemode, then after this screen your game will automatically be open to LAN.
Then, in the chat box your game will tell you the port to join the server, the other person will use this number to join the server.
Here is my friend, Whispeon (Ben Herbert) joining the game, as proof that this has worked properly.
And here he is in the flesh.. or pixels..? But it worked! Have fun with your LAN server and friends.