The Normal budgie is one whose markings match with the wild type budgerigar. The budgie can be any of the basic body colors, but so long as it has the wild type markings it will be of the Normal variety. So you can have Skyblue Normal, Grey Normal, Dark Green Normal.... etc.
Starting with a wild budgie, pictured on the right, the body colour is a lovely bright light green, with a yellow face. Along the bottom edge of the face are six small black spots and sitting over the outer of these on each side is a violet cheek patch. From the forehead back and down to between the wings they have black and yellow stripes. The feathers on the wings are blackish with yellow edging, with the long flight feathers being black-blue with a thick yellow bar only visible when the wings are extended. The long tail feathers are a lovely dark teal blue, with the tail coverts having a thick yellow bar through them. The eyes have a black pupil surrounded by a white circle called the iris ring. The legs and feet are grey. A male budgie can be recognised by the smooth blue cere (flesh around the nostrils), see the bird pictured onthe right. While the female has a brown, usually rough cere.
As mentioned above the normal budgie can come in all the basic body colours; Light Green, Dark Green, Olive, Grey Green, Violet Green, Skyblue, Cobalt, Mauve, Grey and Violet.
It can also be combined with other varieties that alter the colour of the markings, to produce many beautiful birds; Cinnamon, Greywing, Clearwing, Dilute, Fallow, Yellow Face to name a few.
Basically any bird that still has the wild type markings is a normal, so you can have a Normal Cinnamon or a Normal Yellow Face Blue etc. It is usual just to skip using the term Normal and refer to them simply as a Cinnamon or a Yellow Faced Blue for convenience though.
Below we have a Grey Green Normal on the left and a Skyblue Yellow Face Type 2 Normal on the right.
The Normal budgie is the standard by which the inheritance of other colours and varieties are described. If a variety is recessive it means that it is recessive to the Normal form. If it is dominant it means it is dominant to the Normal form.
So to describe how the Normal is inherited you really need to know how the variety it is breeding with works. Take a look at the other varieties to understand how they react with it.